cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119
Offset: 1

Views

Author

Marc LeBrun

Keywords

Comments

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

Examples

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

References

  • Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Links

Crossrefs

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

Programs

  • Haskell
    a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014
  • Maple
    a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017
  • Mathematica
    a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)
  • PARI
    a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))
  • PARI
    a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014
  • Perl
    use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016
  • Python
    from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

Formula

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

A001477 The nonnegative integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - N. J. A. Sloane, Mar 13 2010
The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - N. J. A. Sloane, Jun 16 2010
Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - M. F. Hasler, Nov 20 2013
The number of partitions of 2n into exactly 2 parts. - Colin Barker, Mar 22 2015
The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.- Philippe A.J.G. Chevalier, Dec 29 2015
Partial sums give A000217. - Omar E. Pol, Jul 26 2018
First differences are A000012 (the "all 1's" sequence). - M. F. Hasler, May 30 2020
See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - M. F. Hasler, Nov 09 2021
This is the unique sequence (a(n)) that satisfies the inequality a(n+1) > a(a(n)) for all n in N. This simple and surprising result comes from the 6th problem proposed by Bulgaria during the second day of the 19th IMO (1977) in Belgrade (see link and reference). - Bernard Schott, Jan 25 2023

Examples

			Triangular view:
   0
   1   2
   3   4   5
   6   7   8   9
  10  11  12  13  14
  15  16  17  18  19  20
  21  22  23  24  25  26  27
  28  29  30  31  32  33  34  35
  36  37  38  39  40  41  42  43  44
  45  46  47  48  49  50  51  52  53  54

References

  • Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.

Links

Crossrefs

Cf. A000027 (n>=1).
Cf. A000012 (first differences).
Partial sums of A057427. - Jeremy Gardiner, Sep 08 2002
Cf. A038608 (alternating signs), A001787 (binomial transform).
Cf. A055112.
Cf. Boustrophedon transforms: A231179, A000737.
Cf. A245422.
Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.
When written as an array, the rows/columns are A000217, A000124, A152948, A152950, A145018, A167499, A166136, A167487... and A000096, A034856, A055998, A046691, A052905, A055999... (with appropriate offsets); cf. analogous lists for A000027 in A185787.
Cf. A000290.
Cf. A061579 (transposed matrix / reversed triangle).

Programs

Formula

a(n) = n.
a(0) = 0, a(n) = a(n-1) + 1.
G.f.: x/(1-x)^2.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2*n*(n+1) (A046092), antidiagonal sums are n*(n+1)*(n+2)/2 (A027480). - Ralf Stephan, Oct 17 2004
Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: x*e^x. - Franklin T. Adams-Watters, Sep 11 2005
a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
Alternating partial sums give A001057 = A000217 - 2*(A008794). - Eric Desbiaux, Oct 28 2008
a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - Eric Desbiaux, Nov 15 2009
a(n) = A007966(n)*A007967(n). - Reinhard Zumkeller, Jun 18 2011
a(n) = Sum_{k>=0} A030308(n,k)*2^k. - Philippe Deléham, Oct 20 2011
a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - R. J. Mathar, Jul 20 2012
a(n+1) = det(C(i+1,j), 1 <= i, j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n-1) = floor(n/e^(1/n)) for n > 0. - Richard R. Forberg, Jun 22 2013
a(n) = A000027(n) for all n>0.
a(n) = floor(cot(1/(n+1))). - Clark Kimberling, Oct 08 2014
a(0)=0, a(n>0) = 2*z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z = A130472(n>0); a 1 to 1 correspondence between integers and naturals. - Adriano Caroli, Mar 29 2015
G.f. as triangle: x*(1 + (x^2 - 5*x + 2)*y + x*(2*x - 1)*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 22 2025

A000265 Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 1, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 1, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 75, 19, 77
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

When n > 0 is written as k*2^j with k odd then k = A000265(n) and j = A007814(n), so: when n is written as k*2^j - 1 with k odd then k = A000265(n+1) and j = A007814(n+1), when n > 1 is written as k*2^j + 1 with k odd then k = A000265(n-1) and j = A007814(n-1).
Also denominator of 2^n/n (numerator is A075101(n)). - Reinhard Zumkeller, Sep 01 2002
Slope of line connecting (o, a(o)) where o = (2^k)(n-1) + 1 is 2^k and (by design) starts at (1, 1). - Josh Locker (joshlocker(AT)macfora.com), Apr 17 2004
Numerator of n/2^(n-1). - Alexander Adamchuk, Feb 11 2005
From Marco Matosic, Jun 29 2005: (Start)
"The sequence can be arranged in a table:
1
1 3 1
1 5 3 7 1
1 9 5 11 3 13 7 15 1
1 17 9 19 5 21 11 23 3 25 13 27 7 29 15 31 1
Every new row is the previous row interspaced with the continuation of the odd numbers.
Except for the ones; the terms (t) in each column are t+t+/-s = t_+1. Starting from the center column of threes and working to the left the values of s are given by A000265 and working to the right by A000265." (End)
This is a fractal sequence. The odd-numbered elements give the odd natural numbers. If these elements are removed, the original sequence is recovered. - Kerry Mitchell, Dec 07 2005
2k + 1 is the k-th and largest of the subsequence of k terms separating two successive equal entries in a(n). - Lekraj Beedassy, Dec 30 2005
It's not difficult to show that the sum of the first 2^n terms is (4^n + 2)/3. - Nick Hobson, Jan 14 2005
In the table, for each row, (sum of terms between 3 and 1) - (sum of terms between 1 and 3) = A020988. - Eric Desbiaux, May 27 2009
This sequence appears in the analysis of A160469 and A156769, which resemble the numerator and denominator of the Taylor series for tan(x). - Johannes W. Meijer, May 24 2009
Indices n such that a(n) divides 2^n - 1 are listed in A068563. - Max Alekseyev, Aug 25 2013
From Alexander R. Povolotsky, Dec 17 2014: (Start)
With regard to the tabular presentation described in the comment by Marco Matosic: in his drawing, starting with the 3rd row, the first term in the row, which is equal to 1 (or, alternatively the last term in the row, which is also equal to 1), is not in the actual sequence and is added to the drawing as a fictitious term (for the sake of symmetry); an actual A000265(n) could be considered to be a(j,k) (where j >= 1 is the row number and k>=1 is the column subscript), such that a(j,1) = 1:
1
1 3
1 5 3 7
1 9 5 11 3 13 7 15
1 17 9 19 5 21 11 23 3 25 13 27 7 29 15 31
and so on ... .
The relationship between k and j for each row is 1 <= k <= 2^(j-1). In this corrected tabular representation, Marco's notion that "every new row is the previous row interspaced with the continuation of the odd numbers" remains true. (End)
Partitions natural numbers to the same equivalence classes as A064989. That is, for all i, j: a(i) = a(j) <=> A064989(i) = A064989(j). There are dozens of other such sequences (like A003602) for which this also holds: In general, all sequences for which a(2n) = a(n) and the odd bisection is injective. - Antti Karttunen, Apr 15 2017
From Paul Curtz, Feb 19 2019: (Start)
This sequence is the truncated triangle:
1, 1;
3, 1, 5;
3, 7, 1, 9;
5, 11, 3, 13, 7;
15, 1, 17, 9, 19, 5;
21, 11, 23, 3, 25, 13, 27;
7, 29, 15, 31, 1, 33, 17, 35;
...
The first column is A069834. The second column is A213671. The main diagonal is A236999. The first upper diagonal is A125650 without 0.
c(n) = ((n*(n+1)/2))/A069834 = 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 1, 1, 8, 8, 1, 1, ... for n > 0. n*(n+1)/2 is the rank of A069834. (End)
As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 27 2019
a(n) is also the map n -> A026741(n) applied at least A007814(n) times. - Federico Provvedi, Dec 14 2021

Examples

			G.f. = x + x^2 + 3*x^3 + x^4 + 5*x^5 + 3*x^6 + 7*x^7 + x^8 + 9*x^9 + 5*x^10 + 11*x^11 + ...

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000004, A000010, A000123, A000217, A000225, A000265, A001227.
Cf. A003602, A003961, A006516, A006519, A007814, A008683, A014577.
Cf. A020988, A025480, A026741, A027750, A035263, A038502, A049606.
Cf. A064989, A065330, A068563, A069834, A075101, A111929, A111930.
Cf. A111918, A111919, A111920, A111921, A111922, A111923, A125650.
Cf. A127793, A132739, A132740, A156769, A160469, A182469, A209308.
Cf. A213671, A220466, A236999, A242603.
Cf. A049606 (partial products), A135013 (partial sums), A099545 (mod 4), A326937 (Dirichlet inverse).
Cf. A026741 (map), A001511 (converging steps), A038550 (prime index).
Cf. A195056 (Dgf at s=3).

Programs

  • Haskell
    a000265 = until odd (`div` 2)
-- Reinhard Zumkeller, Jan 08 2013, Apr 08 2011, Oct 14 2010
  • Java
    int A000265(n){
    while(n%2==0) n>>=1;
    return n;
}
/* Aidan Simmons, Feb 24 2019 */
  • Julia
    using IntegerSequences
[OddPart(n) for n in 1:77] |> println  # Peter Luschny, Sep 25 2021
  • Magma
    A000265:= func< n | n/2^Valuation(n,2) >;
[A000265(n): n in [1..120]]; // G. C. Greubel, Jul 31 2024
  • Maple
    A000265:=proc(n) local t1,d; t1:=1; for d from 1 by 2 to n do if n mod d = 0 then t1:=d; fi; od; t1; end: seq(A000265(n), n=1..77);
A000265 := n -> n/2^padic[ordp](n,2): seq(A000265(n), n=1..77); # Peter Luschny, Nov 26 2010
  • Mathematica
    a[n_Integer /; n > 0] := n/2^IntegerExponent[n, 2]; Array[a, 77] (* Josh Locker *)
a[ n_] := If[ n == 0, 0, n / 2^IntegerExponent[ n, 2]]; (* Michael Somos, Dec 17 2014 *)
  • PARI
    {a(n) = n >> valuation(n, 2)}; /* Michael Somos, Aug 09 2006, edited by M. F. Hasler, Dec 18 2014 */
  • Python
    from _future_ import division
def A000265(n):
    while not n % 2:
        n //= 2
    return n # Chai Wah Wu, Mar 25 2018
  • Python
    def a(n):
    while not n&1: n >>= 1
    return n
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Jun 26 2025
  • SageMath
    def A000265(n): return n//2^valuation(n,2)
[A000265(n) for n in (1..121)] # G. C. Greubel, Jul 31 2024
  • Scheme
    (define (A000265 n) (let loop ((n n)) (if (odd? n) n (loop (/ n 2))))) ;; Antti Karttunen, Apr 15 2017

Formula

a(n) = if n is odd then n, otherwise a(n/2). - Reinhard Zumkeller, Sep 01 2002
a(n) = n/A006519(n) = 2*A025480(n-1) + 1.
Multiplicative with a(p^e) = 1 if p = 2, p^e if p > 2. - David W. Wilson, Aug 01 2001
a(n) = Sum_{d divides n and d is odd} phi(d). - Vladeta Jovovic, Dec 04 2002
G.f.: -x/(1 - x) + Sum_{k>=0} (2*x^(2^k)/(1 - 2*x^(2^(k+1)) + x^(2^(k+2)))). - Ralf Stephan, Sep 05 2003
(a(k), a(2k), a(3k), ...) = a(k)*(a(1), a(2), a(3), ...) In general, a(n*m) = a(n)*a(m). - Josh Locker (jlocker(AT)mail.rochester.edu), Oct 04 2005
a(n) = Sum_{k=0..n} A127793(n,k)*floor((k+2)/2) (conjecture). - Paul Barry, Jan 29 2007
Dirichlet g.f.: zeta(s-1)*(2^s - 2)/(2^s - 1). - Ralf Stephan, Jun 18 2007
a(A132739(n)) = A132739(a(n)) = A132740(n). - Reinhard Zumkeller, Aug 27 2007
a(n) = 2*A003602(n) - 1. - Franklin T. Adams-Watters, Jul 02 2009
a(n) = n/gcd(2^n,n). (This also shows that the true offset is 0 and a(0) = 0.) - Peter Luschny, Nov 14 2009
a(-n) = -a(n) for all n in Z. - Michael Somos, Sep 19 2011
From Reinhard Zumkeller, May 01 2012: (Start)
A182469(n, k) = A027750(a(n), k), k = 1..A001227(n).
a(n) = A182469(n, A001227(n)). (End)
a((2*n-1)*2^p) = 2*n - 1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 05 2013
G.f.: G(0)/(1 - 2*x^2 + x^4) - 1/(1 - x), where G(k) = 1 + 1/(1 - x^(2^k)*(1 - 2*x^(2^(k+1)) + x^(2^(k+2)))/(x^(2^k)*(1 - 2*x^(2^(k+1)) + x^(2^(k+2))) + (1 - 2*x^(2^(k+2)) + x^(2^(k+3)))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 06 2013
a(n) = A003961(A064989(n)). - Antti Karttunen, Apr 15 2017
Completely multiplicative with a(2) = 1 and a(p) = p for prime p > 2, i.e., the sequence b(n) = a(n) * A008683(n) for n > 0 is the Dirichlet inverse of a(n). - Werner Schulte, Jul 08 2018
From Peter Bala, Feb 27 2019: (Start)
O.g.f.: F(x) - F(x^2) - F(x^4) - F(x^8) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4) + (1/2)*L(x^8) + ..., where L(x) = log(1/(1 - x)).
Sum_{n >= 1} x^n/a(n) = 1/2*log(G(x)), where G(x) = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + ... is the o.g.f. of A000123. (End)
O.g.f.: Sum_{n >= 1} phi(2*n-1)*x^(2*n-1)/(1 - x^(2*n-1)), where phi(n) is the Euler totient function A000010. - Peter Bala, Mar 22 2019
a(n) = A049606(n) / A049606(n-1). - Flávio V. Fernandes, Dec 08 2020
a(n) = numerator of n/2^(floor(n/2)). - Federico Provvedi, Dec 14 2021
a(n) = Sum_{d divides n} (-1)^(d+1)*phi(2*n/d). - Peter Bala, Jan 14 2024
a(n) = A030101(A030101(n)). - Darío Clavijo, Sep 19 2024

Extensions

Additional comments from Henry Bottomley, Mar 02 2000
More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2000
Name clarified by David A. Corneth, Apr 15 2017

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Least significant bit of n, lsb(n).
Also decimal expansion of 1/99.
Also the binary expansion of 1/3. - Robert G. Wilson v, Sep 01 2015
a(n) = A134451(n) mod 2. - Reinhard Zumkeller, Oct 27 2007 [Corrected by Jianing Song, Nov 22 2019]
Characteristic function of odd numbers: a(A005408(n)) = 1, a(A005843(n)) = 0. - Reinhard Zumkeller, Sep 29 2008
A102370(n) modulo 2. - Philippe Deléham, Apr 04 2009
Base b expansion of 1/(b^2-1) for any b >= 2 is 0.0101... (A005563 has b^2-1). - Rick L. Shepherd, Sep 27 2009
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i] := 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,1). - Milan Janjic, Jan 24 2010
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,
or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,
or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)
Also parity of the nonnegative integers A001477. - Omar E. Pol, Jan 17 2012
a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - Wolfdieter Lang, May 28 2013
Also the inverse binomial transform of A131577. - Paul Curtz, Nov 16 2016 [an observation forwarded by Jean-François Alcover]
The emanation sequence for the globe category. That is take the globe category, take the corresponding polynomial comonad, consider its carrier polynomial as a generating function, and take the corresponding sequence. - David Spivak, Sep 25 2020
For n > 0, a(n) is the alternating sum of the product of n increasing and n decreasing odd factors. For example, a(4) = 1*7 - 3*5 + 5*3 - 7*1 and a(5) = 1*9 - 3*7 + 5*5 - 7*3 + 9*1. - Charlie Marion, Mar 24 2022

Examples

			G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Ones complement of A059841.
Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
Period k zigzag sequences: this sequence (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
Cf. A154955 (Mobius transform), A131577 (binomial transform).
Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).
Parity of A005811.

Programs

Formula

a(n) = (1 - (-1)^n)/2.
a(n) = n mod 2.
a(n) = 1 - a(n-1).
Multiplicative with a(p^e) = p mod 2. - David W. Wilson, Aug 01 2001
G.f.: x/(1-x^2). E.g.f.: sinh(x). - Paul Barry, Mar 11 2003
a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
a(n) = ceiling((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005
Dirichlet g.f.: (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011
a(n) = ceiling(n/2) - floor(n/2). - Arkadiusz Wesolowski, Sep 16 2012
a(n) = ceiling( cos(Pi*(n-1))/2 ). - Wesley Ivan Hurt, Jun 16 2013
a(n) = floor((n-1)/2) - floor((n-2)/2). - Mikael Aaltonen, Feb 26 2015
Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - Ralf Stephan, Mar 27 2015
a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - Natan Arie Consigli, May 02 2015
Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - Michael Somos, Feb 20 2024

A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 21, 5, 34, 23, 53, 6, 59, 29, 20, 1, 33, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16
Offset: 1

Views

Author

Vladeta Jovovic, Oct 30 2001

Keywords

Comments

From Antti Karttunen, May 12 2014: (Start)
a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]
Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.
(End)
From Antti Karttunen, Dec 18-21 2014: (Start)
When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).
For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.
(End)
Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

Examples

			a(20) = a(2^2*5) = a(2^2)*a(5) = prevprime(5) = 3.

Links

Crossrefs

Cf. A064216 (odd bisection), A003961 (inverse), A151799.
Cf. A000040, A008578, A027746, A032742, A049084, A052126, A055396, A061395, A112798, A249817, A249818.
Other sequences whose definition involve or are some other way related with this sequence: A105560, A108951, A118306, A122111, A156552, A163511, A200746, A241909, A243070, A243071, A243072, A243073, A244319, A245605, A245607, A246165, A246266, A246268, A246277, A246278, A246361, A246362, A246371, A246372, A246373, A246374, A246376, A246380, A246675, A246682, A249745, A250470.
Similar prime-shifts towards smaller numbers: A252461, A252462, A252463.

Programs

  • Haskell
    a064989 1 = 1
a064989 n = product $ map (a008578 . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A064989 n) (if (= 1 n) n (apply * (map (lambda (k) (if (zero? k) 1 (A000040 k))) (map -1+ (map A049084 (factor n)))))))
;; Antti Karttunen, May 12 2014
(definec (A064989 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A064989 (A032742 n))))) ;; One based on given recurrence and utilizing memoizing definec-macro.
(definec (A064989 n) (cond ((= 1 n) n) ((even? n) (A064989 (/ n 2))) (else (A163511 (/ (- (A243071 n) 1) 2))))) ;; Corresponds to one of the alternative formulas, but is very unpractical way to compute this sequence. - Antti Karttunen, Dec 18 2014
  • Maple
    q:= proc(p) prevprime(p) end proc: q(2):= 1:
[seq(mul(q(f[1])^f[2], f = ifactors(n)[2]), n = 1 .. 1000)]; # Robert Israel, Dec 21 2014
  • Mathematica
    Table[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, Jan 04 2016 *)
  • PARI
    { for (n=1, 1000, f=factor(n)~; a=1; j=1; if (n>1 && f[1, 1]==2, j=2); for (i=j, length(f), a*=precprime(f[1, i] - 1)^f[2, i]); write("b064989.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009
  • PARI
    a(n) = {my(f = factor(n)); for (i=1, #f~, if ((p=f[i,1]) % 2, f[i,1] = precprime(p-1), f[i,1] = 1);); factorback(f);} \\ Michel Marcus, Dec 18 2014
  • PARI
    A064989(n)=factorback(Mat(apply(t->[max(precprime(t[1]-1),1),t[2]],Vec(factor(n)~))~)) \\ M. F. Hasler, Dec 29 2014
  • Python
    from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017
  • Python
    from math import prod
from sympy import prevprime, factorint
def A064989(n): return prod(prevprime(p)**e for p, e in  factorint(n>>(~n&n-1).bit_length()).items()) # Chai Wah Wu, Jan 05 2023

Formula

From Antti Karttunen, Dec 18 2014: (Start)
If n = product A000040(k)^e(k) then a(n) = product A008578(k)^e(k) [where A000040(n) gives the n-th prime, and A008578(n) gives 1 for 1 and otherwise the (n-1)-th prime].
a(1) = 1; for n > 1, a(n) = A008578(A055396(n)) * a(A032742(n)). [Above formula represented as a recurrence. Cf. A252461.]
a(1) = 1; for n > 1, a(n) = A008578(A061395(n)) * a(A052126(n)). [Compare to the formula of A252462.]
This prime-shift operation is used in the definitions of many other sequences, thus it can be expressed in many alternative ways:
a(n) = A200746(n) / n.
a(n) = A242424(n) / A105560(n).
a(n) = A122111(A122111(n)/A105560(n)) = A122111(A052126(A122111(n))). [In A112798-partition context: conjugate, remove the largest part (the largest prime factor), and conjugate again.]
a(1) = 1; for n > 1, a(2n) = a(n), a(2n+1) = A163511((A243071(2n+1)-1) / 2).
a(n) = A249818(A250470(A249817(n))). [A250470 is an analogous operation for "going one step up" in the square array A083221 (A083140).]
(End)
Product_{k=1..n} a(k) = n! / A307035(n). - Vaclav Kotesovec, Mar 21 2019
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.220703928... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Nov 18 2022

A001511 The ruler function: exponent of the highest power of 2 dividing 2n. Equivalently, the 2-adic valuation of 2n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 1, 3, 1, 2, 1, 4, 1
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of 2's dividing 2*n.
a(n) is equivalently the exponent of the smallest power of 2 which does not divide n. - David James Sycamore, Oct 02 2023
a(n) - 1 is the number of trailing zeros in the binary expansion of n.
If you are counting in binary and the least significant bit is numbered 1, the next bit is 2, etc., a(n) is the bit that is incremented when increasing from n-1 to n. - Jud McCranie, Apr 26 2004
Number of steps to reach an integer starting with (n+1)/2 and using the map x -> x*ceiling(x) (cf. A073524).
a(n) is the number of the disk to be moved at the n-th step of the optimal solution to Towers of Hanoi problem (comment from Andreas M. Hinz).
Shows which bit to flip when creating the binary reflected Gray code (bits are numbered from the right, offset is 1). This is essentially equivalent to Hinz's comment. - Adam Kertesz, Jul 28 2001
a(n) is the Hamming distance between n and n-1 (in binary). This is equivalent to Kertesz's comments above. - Tak-Shing Chan (chan12(AT)alumni.usc.edu), Feb 25 2003
Let S(0) = {1}, S(n) = {S(n-1), S(n-1)-{x}, x+1} where x = last term of S(n-1); sequence gives S(infinity). - Benoit Cloitre, Jun 14 2003
The sum of all terms up to and including the first occurrence of m is 2^m-1. - Donald Sampson (marsquo(AT)hotmail.com), Dec 01 2003
m appears every 2^m terms starting with the 2^(m-1)th term. - Donald Sampson (marsquo(AT)hotmail.com), Dec 08 2003
Sequence read mod 4 gives A092412. - Philippe Deléham, Mar 28 2004
If q = 2n/2^A001511(n) and if b(m) is defined by b(0)=q-1 and b(m)=2*b(m-1)+1, then 2n = b(A001511(n)) + 1. - Gerald McGarvey, Dec 18 2004
Repeating pattern ABACABADABACABAE ... - Jeremy Gardiner, Jan 16 2005
Relation to C(n) = Collatz function iteration using only odd steps: a(n) is the number of right bits set in binary representation of A004767(n) (numbers of the form 4*m+3). So for m=A004767(n) it follows that there are exactly a(n) recursive steps where m
Between every two instances of any positive integer m there are exactly m distinct values (1 through m-1 and one value greater than m). - Franklin T. Adams-Watters, Sep 18 2006
Number of divisors of n of the form 2^k. - Giovanni Teofilatto, Jul 25 2007
Every prefix up to (but not including) the first occurrence of some k >= 2 is a palindrome. - Gary W. Adamson, Sep 24 2008
1 interleaved with (2 interleaved with (3 interleaved with ( ... ))). - Eric D. Burgess (ericdb(AT)gmail.com), Oct 17 2009
A054525 (Möbius transform) * A001511 = A036987 = A047999^(-1) * A001511. - Gary W. Adamson, Oct 26 2009
Equals A051731 * A036987, (inverse Möbius transform of the Fredholm-Rueppel sequence) = A047999 * A036987. - Gary W. Adamson, Oct 26 2009
Cf. A173238, showing links between generalized ruler functions and A000041. - Gary W. Adamson, Feb 14 2010
Given A000041, P(x) = A(x)/A(x^2) with P(x) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), A(x) = (1 + x + 3x^2 + 4x^3 + 10x^4 + 13x^5 + ...), A(x^2) = (1 + x^2 + 3x^4 + 4x^6 + 10x^8 + ...), where A092119 = (1, 1, 3, 4, 10, ...) = Euler transform of the ruler sequence, A001511. - Gary W. Adamson, Feb 11 2010
Subtracting 1 from every term and deleting any 0's yields the same sequence, A001511. - Ben Branman, Dec 28 2011
In the listing of the compositions of n as lists in lexicographic order, a(k) is the last part of composition(k) for all k <= 2^(n-1) and all n, see example. - Joerg Arndt, Nov 12 2012
According to Hinz, et al. (see links), this sequence was studied by Louis Gros in his 1872 pamphlet "Théorie du Baguenodier" and has therefore been called the Gros sequence.
First n terms comprise least squarefree word of length n using positive integers, where "squarefree" means that the word contains no consecutive identical subwords; e.g., 1 contains no square; 11 contains a square but 12 does not; 121 contains no square; both 1211 and 1212 have squares but 1213 does not; etc. - Clark Kimberling, Sep 05 2013
Length of 0-run starting from 2 (10, 100, 110, 1000, 1010, ...), or length of 1-run starting from 1 (1, 11, 101, 111, 1001, 1011, ...) of every second number, from right to left in binary representation. - Armands Strazds, Apr 13 2017
a(n) is also the frequency of the largest part in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017
As A000005(n) equals the number of even divisors of 2n and A001227(n) = A001227(2n), the formula A001511(n) = A000005(n)/A001227(n) might be read as "The number of even divisors of 2n is always divisible by the number of odd divisors of 2n" (where number of divisors means sum of zeroth powers of divisors). Conjecture: For any nonnegative integer k, the sum of the k-th powers of even divisors of n is always divisible by the sum of the k-th powers of odd divisors of n. - Ivan N. Ianakiev, Jul 06 2019
From Benoit Cloitre, Jul 14 2022: (Start)
To construct the sequence, start from 1's separated by a place 1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,,1,...
Then put the 2's in every other remaining place
1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,,1,2,1,...
Then the 3's in every other remaining place
1,2,1,3,1,2,1,,1,2,1,3,1,2,1,,1,2,1,3,1,2,1,,1,2,1,...
Then the 4's in every other remaining place
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,,1,2,1,3,1,2,1,4,1,2,1,...
By iterating this process, we get the ruler function 1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,... (End)
a(n) is the least positive integer k for which there does not exist i+j=n and a(i)=a(j)=k (cf. A322523). - Rémy Sigrist and Jianing Song, Aug 23 2022
a(n) is the smallest positive integer that does not occur in the coincidences of the sequence so far a(1..n-1) and its reverse. - Neal Gersh Tolunsky, Jan 18 2023
The geometric mean of this sequence approaches the Somos constant (A112302). - Jwalin Bhatt, Jan 31 2025

Examples

			For example, 2^1|2, 2^2|4, 2^1|6, 2^3|8, 2^1|10, 2^2|12, ... giving the initial terms 1, 2, 1, 3, 1, 2, ...
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
1;
2,1;
3,1,2,1;
4,1,2,1,3,1,2,1;
5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
6,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1;
7,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,...
(End)
S(0) = {} S(1) = 1 S(2) = 1, 2, 1 S(3) = 1, 2, 1, 3, 1, 2, 1 S(4) = 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1. - Yann David (yann_david(AT)hotmail.com), Mar 21 2010
From _Joerg Arndt_, Nov 12 2012: (Start)
The 16 compositions of 5 as lists in lexicographic order:
[ n]  a(n)  composition
[ 1]  [ 1]  [ 1 1 1 1 1 ]
[ 2]  [ 2]  [ 1 1 1 2 ]
[ 3]  [ 1]  [ 1 1 2 1 ]
[ 4]  [ 3]  [ 1 1 3 ]
[ 5]  [ 1]  [ 1 2 1 1 ]
[ 6]  [ 2]  [ 1 2 2 ]
[ 7]  [ 1]  [ 1 3 1 ]
[ 8]  [ 4]  [ 1 4 ]
[ 9]  [ 1]  [ 2 1 1 1 ]
[10]  [ 2]  [ 2 1 2 ]
[11]  [ 1]  [ 2 2 1 ]
[12]  [ 3]  [ 2 3 ]
[13]  [ 1]  [ 3 1 1 ]
[14]  [ 2]  [ 3 2 ]
[15]  [ 1]  [ 4 1 ]
[16]  [ 5]  [ 5 ]
a(n) is the last part in each list.
(End)
From _Omar E. Pol_, Aug 20 2013: (Start)
Also written as a triangle in which the right border gives A000027 and row lengths give A011782 and row sums give A000079 the sequence begins:
1;
2;
1,3;
1,2,1,4;
1,2,1,3,1,2,1,5;
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6;
1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7;
(End)
G.f. = x + 2*x^2 + x^3 + 3*x^4 + x^5 + 2*x^6 + x^7 + 4*x^8 + x^9 + 2*x^10 + ...

References

  • J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
  • E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 2nd ed., 2001-2003; see Dim- and Dim+ on p. 98; Dividing Rulers, on pp. 436-437; The Ruler Game, pp. 469-470; Ruler Fours, Fives, ... Fifteens on p. 470.
  • L. Gros, Théorie du Baguenodier, Aimé Vingtrinier, Lyon, 1872.
  • R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E22.
  • A. M. Hinz, The Tower of Hanoi, in Algebras and combinatorics (Hong Kong, 1997), 277-289, Springer, Singapore, 1999.
  • D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.1.3, Problem 41, p. 589.
  • Andrew Schloss, "Towers of Hanoi" composition, in The Digital Domain. Elektra/Asylum Records 9 60303-2, 1983. Works by Jaffe (Finale to "Silicon Valley Breakdown"), McNabb ("Love in the Asylum"), Schloss ("Towers of Hanoi"), Mattox ("Shaman"), Rush, Moorer ("Lions are Growing") and others.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 1 of table A050600.
Sequence read mod 2 gives A035263.
Sequence is bisection of A007814, A050603, A050604, A067029, A089309.
This is Guy Steele's sequence GS(4, 2) (see A135416).
Cf. A005187 (partial sums), A085058 (bisection), A112302 (geometric mean), A171977 (2^a(n)).
Cf. A000005, A000041, A000079, A003188, A003278, A003602, A007949, A018238, A047999, A051731, A054525, A054852, A065176, A082850, A089080, A092119, A117303, A129360, A130093, A173238, A181988, A220466.
Cf. A287896, A002487, A209229 (Mobius trans.), A092673 (Dirichlet inv.).
Cf. generalized ruler functions for k=3,4,5: A051064, A115362, A055457.

Programs

  • Haskell
    a001511 n = length $ takeWhile ((== 0) . (mod n)) a000079_list
-- Reinhard Zumkeller, Sep 27 2011
  • Haskell
    a001511 n | odd n = 1 | otherwise = 1 + a001511 (n `div` 2)
-- Walt Rorie-Baety, Mar 22 2013
  • MATLAB
    nmax=5;r=1;for n=2:nmax;r=[r n r];end % Adriano Caroli, Feb 26 2016
  • Magma
    [Valuation(2*n,2): n in [1..105]]; // Bruno Berselli, Nov 23 2015
  • Maple
    A001511 := n->2-wt(n)+wt(n-1); # where wt is defined in A000120
# This is the binary logarithm of the denominator of (256^n-1)B_{8n}/n, in Maple parlance a := n -> log[2](denom((256^n-1)*bernoulli(8*n)/n)). - Peter Luschny, May 31 2009
A001511 := n -> padic[ordp](2*n,2): seq(A001511(n), n=1..105);  # Peter Luschny, Nov 26 2010
a:= n-> ilog2((Bits[Xor](2*n, 2*n-1)+1)/2): seq(a(n), n=1..50);  # Gary Detlefs, Dec 13 2018
  • Mathematica
    Array[ If[ Mod[ #, 2] == 0, FactorInteger[ # ][[1, 2]], 0] &, 105] + 1 (* or *)
Nest[ Flatten[ # /. a_Integer -> {1, a + 1}] &, {1}, 7] (* Robert G. Wilson v, Mar 04 2005 *)
IntegerExponent[2*n, 2] (* Alexander R. Povolotsky, Aug 19 2011 *)
myHammingDistance[n_, m_] := Module[{g = Max[m, n], h = Min[m, n]}, b1 = IntegerDigits[g, 2]; b2 = IntegerDigits[h, 2, Length[b1]]; HammingDistance[b1, b2]] (* Vladimir Shevelev A206853 *) Table[ myHammingDistance[n, n - 1], {n, 111}] (* Robert G. Wilson v, Apr 05 2012 *)
Table[Position[Reverse[IntegerDigits[n,2]],1,1,1],{n,110}]//Flatten (* Harvey P. Dale, Aug 18 2017 *)
  • PARI
    a(n) = sum(k=0,floor(log(n)/log(2)),floor(n/2^k)-floor((n-1)/2^k)) /* Ralf Stephan */
  • PARI
    a(n)=if(n%2,1,factor(n)[1,2]+1) /* Jon Perry, Jun 06 2004 */
  • PARI
    {a(n) = if( n, valuation(n, 2) + 1, 0)}; /* Michael Somos, Sep 30 2006 */
  • PARI
    {a(n)=if(n==1,1,polcoeff(x-sum(k=1, n-1, a(k)*x^k*(1-x^k)*(1-x+x*O(x^n))), n))} /* Paul D. Hanna, Jun 22 2007 */
  • Python
    def a(n): return bin(n)[2:][::-1].index("1") + 1 # Indranil Ghosh, May 11 2017
  • Python
    A001511 = lambda n: (n&-n).bit_length() # M. F. Hasler, Apr 09 2020
  • Python
    def A001511(n): return (~n & n-1).bit_length()+1 # Chai Wah Wu, Jul 01 2022
  • Sage
    [valuation(2*n,2) for n in (1..105)]  # Bruno Berselli, Nov 23 2015
  • Scheme
    (define (A001511 n) (let loop ((n n) (e 1)) (if (odd? n) e (loop (/ n 2) (+ 1 e))))) ;; Antti Karttunen, Oct 06 2017

Formula

a(n) = A007814(n) + 1 = A007814(2*n).
a(2*n+1) = 1; a(2*n) = 1 + a(n). - Philippe Deléham, Dec 08 2003
a(n) = 2 - A000120(n) + A000120(n-1), n >= 1. - Daniele Parisse
a(n) = 1 + log_2(abs(A003188(n) - A003188(n-1))).
Multiplicative with a(p^e) = e+1 if p = 2; 1 if p > 2. - David W. Wilson, Aug 01 2001
For any real x > 1/2: lim_{N->infinity} (1/N)*Sum_{n=1..N} x^(-a(n)) = 1/(2*x-1); also lim_{N->infinity} (1/N)*Sum_{n=1..N} 1/a(n) = log(2). - Benoit Cloitre, Nov 16 2001
s(n) = Sum_{k=1..n} a(k) is asymptotic to 2*n since s(n) = 2*n - A000120(n). - Benoit Cloitre, Aug 31 2002
For any n >= 0, for any m >= 1, a(2^m*n + 2^(m-1)) = m. - Benoit Cloitre, Nov 24 2002
a(n) = Sum_{d divides n and d is odd} mu(d)*tau(n/d). - Vladeta Jovovic, Dec 04 2002
G.f.: A(x) = Sum_{k>=0} x^(2^k)/(1-x^(2^k)). - Ralf Stephan, Dec 24 2002
a(1) = 1; for n > 1, a(n) = a(n-1) + (-1)^n*a(floor(n/2)). - Vladeta Jovovic, Apr 25 2003
A fixed point of the mapping 1->12; 2->13; 3->14; 4->15; 5->16; ... . - Philippe Deléham, Dec 13 2003
Product_{k>0} (1+x^k)^a(k) is g.f. for A000041(). - Vladeta Jovovic, Mar 26 2004
G.f. A(x) satisfies A(x) = A(x^2) + x/(1-x). - Franklin T. Adams-Watters, Feb 09 2006
a(A118413(n,k)) = A002260(n,k); = a(A118416(n,k)) = A002024(n,k); a(A014480(n)) = A003602(A014480(n)). - Reinhard Zumkeller, Apr 27 2006
Ordinal transform of A003602. - Franklin T. Adams-Watters, Aug 28 2006 (The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.)
Could be extended to n <= 0 using a(-n) = a(n), a(0) = 0, a(2*n) = a(n)+1 unless n=0. - Michael Somos, Sep 30 2006
A094267(2*n) = A050603(2*n) = A050603(2*n + 1) = a(n). - Michael Somos, Sep 30 2006
Sequence = A129360 * A000005 = M*V, where M = an infinite lower triangular matrix and V = d(n) as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Row sums of triangle A130093. - Gary W. Adamson, May 13 2007
Dirichlet g.f.: zeta(s)*2^s/(2^s-1). - Ralf Stephan, Jun 17 2007
a(n) = -Sum_{d divides n} mu(2*d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n*( 1 - x^n ). - Paul D. Hanna, Jun 22 2007
2*n = 2^a(n)* A000265(n). - Eric Desbiaux, May 14 2009 [corrected by Alejandro Erickson, Apr 17 2012]
Multiplicative with a(2^k) = k + 1, a(p^k) = 1 for any odd prime p. - Franklin T. Adams-Watters, Jun 09 2009
With S(n): 2^n - 1 first elements of the sequence then S(0) = {} (empty list) and if n > 0, S(n) = S(n-1), n, S(n-1). - Yann David (yann_david(AT)hotmail.com), Mar 21 2010
a(n) = log_2(A046161(n)/A046161(n-1)). - Johannes W. Meijer, Nov 04 2012
a((2*n-1)*2^p) = p+1, p >= 0 and n >= 1. - Johannes W. Meijer, Feb 05 2013
a(n+1) = 1 + Sum_{j=0..ceiling(log_2(n+1))} (j * (1 - abs(sign((n mod 2^(j + 1)) - 2^j + 1)))). - Enrico Borba, Oct 01 2015
Conjecture: a(n) = A181988(n)/A003602(n). - L. Edson Jeffery, Nov 21 2015
a(n) = log_2(A006519(n)) + 1. - Doug Bell, Jun 02 2017
Inverse Moebius transform of A209229. - Andrew Howroyd, Aug 04 2018
a(n) = 1 + (A183063(n)/A001227(n)). - Omar E. Pol, Nov 06 2018 (after Franklin T. Adams-Watters)
a(n) = log_2((Xor(2*n,2*n-1)+1)/2). - Gary Detlefs, Dec 13 2018
(2^(a(n)-1)-1)*(n mod 4) = 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Dec 14 2018
a(n) = A000005(n)/A001227(n). - Ivan N. Ianakiev, Jul 05 2019
a(n) = Sum_{j=1..r} (j/2^j)*(Product_{k=1..j} (1 - (-1)^floor( (n+2^(j-1))/2^(k-1) ))), for n < a predefined 2^r. - Adriano Caroli, Sep 30 2019

Extensions

Name edited following suggestion by David James Sycamore, Oct 05 2023

A001227 Number of odd divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 8
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - Tom Verhoeff
Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - Henry Bottomley, Apr 13 2000
This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - Gus Wiseman, Oct 04 2022
a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003
a(n) = 1 if and only if n is a power of 2 (see A000079). - Lekraj Beedassy, Apr 12 2005
Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005
For n odd, n is prime if and only if a(n) = 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006
Lengths of rows of triangle A182469;
Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013
Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014
Row sums of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
Row sums of triangle A237048. - Omar E. Pol, Oct 24 2014
A069288(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - Juri-Stepan Gerasimov, May 23 2016, Jul 15 2016
Also the number of odd divisors of n*2^m for m >= 0. - Juri-Stepan Gerasimov, Jul 15 2016
a(n) is odd if and only if n is a square or twice a square. - Juri-Stepan Gerasimov, Jul 17 2016
a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - Omar E. Pol, Nov 05 2016
a(n) is also the number of partitions of n into an odd number of equal parts. - Omar E. Pol, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - N. J. A. Sloane, Dec 03 2020]

Examples

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 + ...
From _Omar E. Pol_, Nov 30 2020: (Start)
For n = 9 there are three odd divisors of 9; they are [1, 3, 9]. On the other hand there are three partitions of 9 into consecutive parts: they are [9], [5, 4] and [4, 3, 2], so a(9) = 3.
Illustration of initial terms:
                              Diagram
   n   a(n)                         _
   1     1                        _|1|
   2     1                      _|1 _|
   3     2                    _|1  |1|
   4     1                  _|1   _| |
   5     2                _|1    |1 _|
   6     2              _|1     _| |1|
   7     2            _|1      |1  | |
   8     1          _|1       _|  _| |
   9     3        _|1        |1  |1 _|
  10     2      _|1         _|   | |1|
  11     2    _|1          |1   _| | |
  12     2   |1            |   |1  | |
...
a(n) is the number of horizontal line segments in the n-th level of the diagram. For more information see A286001. (End)

References

  • B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.
  • J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).
  • Ronald. L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
  • P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

Links

Crossrefs

Cf. A000005, A000079, A000593, A010054 (char. func.), A038547 (positions of first appearances), A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A237593, A247795, A272887, A273401, A279387, A286001.
Cf. A000203, A001620, A002162, A053866, A060831.
If this sequence counts gapless sets by sum (by Sylvester's enumeration), these sets are ranked by A073485 and A356956. See also A055932, A066311, A073491, A107428, A137921, A333217, A356224, A356841, A356845.
Dirichlet inverse is A327276.

Programs

  • Haskell
    a001227 = sum . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, May 01 2012, Jul 25 2011
  • Magma
    [NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // Vincenzo Librandi, Jun 02 2019
  • Maple
    for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:
A001227 := proc(n) local a,d;
    a := 1 ;
    for d in ifactors(n)[2] do
        if op(1,d) > 2 then
            a := a*(op(2,d)+1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 18 2015
  • Mathematica
    f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 27 2004 *)
Table[Total[Mod[Divisors[n], 2]],{n,105}] (* Zak Seidov, Apr 16 2010 *)
f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}]; (* Michael Somos, May 17 2013 *)
a[ n_] := DivisorSum[ n, Mod[ #, 2] &]; (* Michael Somos, May 17 2013 *)
Count[Divisors[#],?OddQ]&/@Range[110] (* _Harvey P. Dale, Feb 15 2015 *)
(* using a262045 from A262045 to compute a(n) = number of subparts in the symmetric representation of sigma(n) *)
(* cl = current level, cs = current subparts count *)
a001227[n_] := Module[{cs=0, cl=0, i, wL, k}, wL=a262045[n]; k=Length[wL]; For[i=1, i<=k, i++, If[wL[[i]]>cl, cs++; cl++]; If[wL[[i]]Hartmut F. W. Hoft, Dec 16 2016 *)
a[n_] := DivisorSigma[0, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)
  • PARI
    {a(n) = sumdiv(n, d, d%2)}; /* Michael Somos, Oct 06 2007 */
  • PARI
    {a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]}; /* Michael Somos, Oct 06 2007 */
  • PARI
    a(n)=numdiv(n>>valuation(n,2)) \\ Charles R Greathouse IV, Mar 16 2011
  • PARI
    a(n)=sum(k=1,round(solve(x=1,n,x*(x+1)/2-n)),(k^2-k+2*n)%(2*k)==0) \\ Charles R Greathouse IV, May 31 2013
  • PARI
    a(n)=sumdivmult(n,d,d%2) \\ Charles R Greathouse IV, Aug 29 2013
  • Python
    from functools import reduce
from operator import mul
from sympy import factorint
def A001227(n): return reduce(mul,(q+1 for p, q in factorint(n).items() if p > 2),1) # Chai Wah Wu, Mar 08 2021
  • SageMath
    def A001227(n): return len([1 for d in divisors(n) if is_odd(d)])
[A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

Formula

Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
Comment from N. J. A. Sloane, Dec 02 2020: (Start)
By counting the odd divisors f n in different ways, we get three different ways of writing the ordinary generating function. It is:
A(x) = x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + ...
= Sum_{k >= 1} x^(2*k-1)/(1-x^(2*k-1))
= Sum_{k >= 1} x^k/(1-x^(2*k))
= Sum_{k >= 1} x^(k*(k+1)/2)/(1-x^k) [Ramanujan, 2nd notebook, p. 355.].
(This incorporates comments from Vladeta Jovovic, Oct 16 2002 and Michael Somos, Oct 30 2005.) (End)
G.f.: x/(1-x) + Sum_{n>=1} x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = Sum_{n>=1} x^n/(1-x^n). - Joerg Arndt, Nov 06 2010
a(n) = A000005(n)/(A007814(n)+1) = A000005(n)/A001511(n).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = A000005(A000265(n)). - Lekraj Beedassy, Jan 07 2005
Moebius transform is period 2 sequence [1, 0, ...] = A000035, which means a(n) is the Dirichlet convolution of A000035 and A057427.
a(n) = A113414(2*n). - N. J. A. Sloane, Jan 24 2006 (corrected Nov 10 2007)
a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller, Apr 18 2006
Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson, Nov 06 2007
a(n) = A000005(n) - A183063(n).
a(n) = d(n) if n is odd, or d(n) - d(n/2) if n is even, where d(n) is the number of divisors of n (A000005). (See the Weisstein page.) - Gary W. Adamson, Mar 15 2011
Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - R. J. Mathar, Jun 28 2011
a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. - Reinhard Zumkeller, May 01 2012
a(n) = 1 + A069283(n). - R. J. Mathar, Jun 18 2015
a(A002110(n)/2) = n, n >= 1. - Altug Alkan, Sep 29 2015
a(n*2^m) = a(n*2^i), a((2*j+1)^n) = n+1 for m >= 0, i >= 0 and j >= 0. a((2*x+1)^n) = a((2*y+1)^n) for positive x and y. - Juri-Stepan Gerasimov, Jul 17 2016
Conjectures: a(n) = A067742(n) + 2*A131576(n) = A082647(n) + A131576(n). - Omar E. Pol, Feb 15 2017
a(n) = A000005(2n) - A000005(n) = A099777(n)-A000005(n). - Danny Rorabaugh, Oct 03 2017
L.g.f.: -log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
G.f.: (psi_{q^2}(1/2) + log(1-q^2))/log(q), where psi_q(z) is the q-digamma function. - Michael Somos, Jun 01 2019
a(n) = A003056(n) - A238005(n). - Omar E. Pol, Sep 12 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023
a(n) = Sum_{i=1..n} (-1)^(i+1)*A135539(n,i). - Ridouane Oudra, Apr 13 2023

A003963 Fully multiplicative with a(p) = k if p is the k-th prime.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 6, 1, 7, 4, 8, 3, 8, 5, 9, 2, 9, 6, 8, 4, 10, 6, 11, 1, 10, 7, 12, 4, 12, 8, 12, 3, 13, 8, 14, 5, 12, 9, 15, 2, 16, 9, 14, 6, 16, 8, 15, 4, 16, 10, 17, 6, 18, 11, 16, 1, 18, 10, 19, 7, 18, 12, 20, 4, 21, 12, 18, 8, 20, 12, 22, 3, 16, 13, 23, 8, 21, 14, 20, 5
Offset: 1

Views

Author

Marc LeBrun

Keywords

Comments

a(n) is the Matula number of the rooted tree obtained from the rooted tree T having Matula number n, by contracting its edges that emanate from the root. Example: a(49) = 16. Indeed, the rooted tree with Matula number 49 is the tree obtained by merging two copies of the tree Y at their roots. Contracting the two edges that emanate from the root, we obtain the star tree with 4 edges having Matula number 16. - Emeric Deutsch, May 01 2015
The Matula (or Matula-Goebel) number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, May 01 2015
a(n) is the product of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(75) = 18; indeed, the partition having Heinz number 75 = 3*5*5 is [2,3,3] and 2*3*3 = 18. - Emeric Deutsch, Jun 03 2015
Let T be the free-commutative-monoid monad on the category Set. Then for each set N we have a canonical function m from TTN to TN. If we let N = {1, 2, 3, ...} and enumerate the primes in the usual way (A000040) then unique prime factorization gives a canonical bijection f from N to TN. Then the sequence is given by a(n) = f^-1(m(T(f)(f(n)))). - Oscar Cunningham, Jul 18 2019

Links

Crossrefs

Cf. A000720, A001221, A001222, A027748, A049084, A056239, A064553, A124010, A156552, A215366, A227184, A241909, A243354, A243499, A243504.
Product of entries on row n of A112798.

Programs

  • Haskell
    a003963 n = product $
   zipWith (^) (map a049084 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 30 2012
  • Maple
    with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then pi(n) else a(r(n))*a(s(n)) end if end proc: seq(a(n), n = 1 .. 88);
# Alternative:
seq(mul(numtheory:-pi(t[1])^t[2], t=ifactors(n)[2]), n=1..100); # Robert Israel, May 01 2015
  • Mathematica
    a[n_] := Times @@ (PrimePi[ #[[1]] ]^#[[2]]& /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 88}]
  • PARI
    a(n)=f=factor(n);prod(i=1,#f[,1],primepi(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Apr 26 2012; corrected by Rémy Sigrist, Jul 18 2019
  • PARI
    a(n) = {f = factor(n); for (i=1, #f~, f[i, 1] = primepi(f[i, 1]); ); factorback(f); } \\ Michel Marcus, Feb 08 2015
  • PARI
    A003963(n)={n=factor(n); n[,1]=apply(primepi,n[,1]); factorback(n)} \\ M. F. Hasler, May 03 2018
  • Python
    from math import prod
from sympy import primepi, factorint
def A003963(n): return prod(primepi(p)**e for p, e in factorint(n).items()) # Chai Wah Wu, Nov 17 2022

Formula

If n = product prime(k)^e(k) then a(n) = product k^e(k).
Multiplicative with a(p^e) = A000720(p)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A049084(A027748(n,k))^A124010(n,k). - Reinhard Zumkeller, Jun 30 2012
Rec. eq.: a(1)=1, a(k-th prime) = a(k), a(rs)=a(r)a(s). The Maple program is based on this. - Emeric Deutsch, May 01 2015
a(n) = A243504(A241909(n)) = A243499(A156552(n)) = A227184(A243354(n)) - Antti Karttunen, Mar 07 2017

A001157 a(n) = sigma_2(n): sum of squares of divisors of n.

Original entry on oeis.org

1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122, 210, 170, 250, 260, 341, 290, 455, 362, 546, 500, 610, 530, 850, 651, 850, 820, 1050, 842, 1300, 962, 1365, 1220, 1450, 1300, 1911, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2562, 2366, 2650, 2210, 3410, 2451, 3255
Offset: 1

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
sigma_2(n) is the sum of the squares of the divisors of n.
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001.
Row sums of triangles A134575 and A134559. - Gary W. Adamson, Nov 02 2007
Also sum of square divisors of n^2. - Michel Marcus, Jan 14 2014
Conjecture: For each k = 2,3,..., all the rational numbers sigma_k(n)/n^k = Sum_{d|n} 1/d^k (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015
5 is the only prime entry in the sequence. - Drake Thomas, Dec 18 2016
4*a(n) = sum of squares of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
  • D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 11.
  • P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table I. The entry 53 should be 50. - N. J. A. Sloane, May 21 2014
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000005, A000203, A001158, A001159, A053807, A064602, A127093, A134841, A048250.
Cf. also A192794, A082063 (gcd(a(n),n) and its largest prime factor); A179931, A192795 (gcd(a(n),A000203(n)) and largest prime factor).
Main diagonal of the array in A242639.
Cf. A333972 (Dgf at s=4).

Programs

  • Haskell
    a001157 n = s n 1 1 a000040_list where
   s 1 1 y _          = y
   s m x y ps'@(p:ps)
     | m `mod` p == 0 = s (m `div` p) (x * p^2) y ps'
     | x > 1          = s m 1 (y * (x * p^2 - 1) `div` (p^2 - 1)) ps
     | otherwise      = s m 1 y ps
-- Reinhard Zumkeller, Jul 10 2011
  • Magma
    [DivisorSigma(2,n): n in [1..50]]; // Bruno Berselli, Apr 10 2013
  • Maple
    with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];
  • Mathematica
    Table[DivisorSigma[2, n], {n, 1, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
DivisorSigma[2,Range[50]] (* Harvey P. Dale, Aug 22 2016 *)
  • Maxima
    makelist(divsum(n,2),n,1,20); /* Emanuele Munarini, Mar 26 2011 */
  • PARI
    a(n)=if(n<1,0,sigma(n,2))
  • PARI
    a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p^2*X))[n])
  • PARI
    a(n)=if(n<1,0,n*polcoeff(sum(k=1,n,x^k/(x^k-1)^2/k,x*O(x^n)),n)) /* Michael Somos, Jan 29 2005 */
  • PARI
    N=99; q='q+O('q^N); Vec(sum(n=1,N,n^2*q^n/(1-q^n)))  /* Joerg Arndt, Feb 04 2011 */
  • PARI
    a(n) = sumdiv(n^2, d, issquare(d)*d); \\ Michel Marcus, Jan 14 2014
  • Python
    from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jan 05 2021
  • Python
    from math import prod
from sympy import factorint
def a(n): return prod((p**(2*e+2)-1)//(p**2-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024
  • Sage
    [sigma(n,2)for n in range(1,51)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: Sum_{k>0} k^2 x^k/(1-x^k). Dirichlet g.f.: zeta(s)*zeta(s-2). - Michael Somos, Apr 05 2003
Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1). - David W. Wilson, Aug 01 2001
G.f. for sigma_k(n): Sum_{m>0} m^k*x^m/(1-x^m). - Vladeta Jovovic, Oct 18 2002
L.g.f.: -log(Product_{j>=1} (1-x^j)^j) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Equals A127093 * [1, 2, 3, ...]. - Gary W. Adamson, May 10 2007
Equals A051731 * [1, 4, 9, 16, 25, ...]. A051731 * [1/1, 1/2, 1/3, 1/4, ...] = [1/1, 5/4, 10/9, 21/16, 26/25, ...]. - Gary W. Adamson, Nov 02 2007
Row sums of triangle A134841. - Gary W. Adamson, Nov 12 2007
a(n) = A035316(n^2). - Michel Marcus, Jan 14 2014
Conjecture: a(n) = sigma(n^2*rad(n))/sigma(rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
G.f.: Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
a(n) = a(n/4) + A050461(n) + A076577(n/2) + A050465(n) where A(.) are zero for non-integer arguments. - R. J. Mathar, May 25 2020
Sum_{k>=1} 1/a(k) = A109694 = 1.53781289182725616253866100273826833091936004947322354929617689659426330445... - Vaclav Kotesovec, Sep 26 2020
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 - ((n-1)^2 - 2)*q^n - ((n+1)^2 - 2)*q^(2*n) + n^2*q^(3*n))/(1 - q^n)^3 - apply the operator x*d/dx twice to equation 5 in Arndt and set x = 1. - Peter Bala, Jan 21 2021
From Vaclav Kotesovec, Aug 07 2022: (Start)
Sum_{k=1..n} a(k) = A064602(n) ~ zeta(3) * n^3 / 3.
Sum_{k=1..n} (-1)^k * a(k) ~ zeta(3) * n^3 / 24. (End)
a(n) = Sum_{1 <= i, j <= n} tau(gcd(i, j, n)) = Sum_{d divides n} tau(d) * J_2(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024

A000583 Fourth powers: a(n) = n^4.

Original entry on oeis.org

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1048576, 1185921
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Figurate numbers based on 4-dimensional regular convex polytope called the 4-measure polytope, 4-hypercube or tesseract with Schlaefli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004
Totally multiplicative sequence with a(p) = p^4 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A058649. The inverse binomial transforms yields the (finite) 0, 1, 14, 36, 24, the 4th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Generate Pythagorean triangles with parameters a and b to get sides of lengths x = b^2-a^2, y = 2*a*b, and z = a^2 + b^2. In particular use a=n-1 and b=n for a triangle with sides (x1,y1,z1) and a=n and b=n+1 for another triangle with sides (x2,y2,z2). Then x1*x2 + y1*y2 + z1*z2 = 8*a(n). - J. M. Bergot, Jul 22 2013
For n > 0, a(n) is the largest integer k such that k^4 + n is a multiple of k + n. Also, for n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n^2. - Derek Orr, Sep 04 2014
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
a(n+2)/2 is the area of a trapezoid with vertices at (T(n), T(n+1)), (T(n+1), T(n)), (T(n+1), T(n+2)), and (T(n+2), T(n+1)) with T(n)=A000292(n) for n >= 0. - J. M. Bergot, Feb 16 2018

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
  • Dov Juzuk, Curiosa 56: An interesting observation, Scripta Mathematica 6 (1939), 218.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 47.

Links

Crossrefs

Cf. A000538, A005917 (first differences), A000332, A014820, A092181, A092182, A092183.
Cf. A004831, A002646.
Cf. A002593, A260810. - Bruno Berselli, Jul 31 2015
Cf. A002415, A000290, A006008, A132366, A039623, A139584, A071270, A047928, A187756.
Cf. A062392, A231303, A267315.

Programs

Formula

a(n) = A123865(n)+1 = A002523(n)-1.
Multiplicative with a(p^e) = p^(4e). - David W. Wilson, Aug 01 2001
G.f.: x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. More generally, g.f. for n^m is Euler(m, x)/(1-x)^(m+1), where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292).
Dirichlet generating function: zeta(s-4). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: (x + 7*x^2 + 6*x^3 + x^4)*e^x. More generally, the general form for the e.g.f. for n^m is phi_m(x)*e^x, where phi_m is the exponential polynomial of order n. - Franklin T. Adams-Watters, Sep 11 2005
Sum_{k>0} 1/a(k) = Pi^4/90 = A013662. - Jaume Oliver Lafont, Sep 20 2009
a(n) = C(n+3,4) + 11*C(n+2,4) + 11*C(n+1,4) + C(n,4). [Worpitzky's identity for powers of 4. See, e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]
a(n) = n*A177342(n) - Sum_{i=1..n-1} A177342(i) - (n - 1), with n > 1. - Bruno Berselli, May 07 2010
a(n) + a(n+1) + 1 = 2*A002061(n+1)^2. - Charlie Marion, Jun 13 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. - Ant King, Sep 23 2013
From Amiram Eldar, Jan 20 2021: (Start)
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/720 (A267315).
Product_{n>=2} (1 - 1/a(n)) = sinh(Pi)/(4*Pi). (End)
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