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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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A007306 Denominators of Farey tree fractions (i.e., the Stern-Brocot subtree in the range [0,1]).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 7, 8, 7, 5, 6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 6, 7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 11, 16, 19, 17, 18, 21, 19, 14, 13, 17, 18, 15, 13, 14, 11, 7, 8, 13, 17, 16, 19, 23, 22, 17, 19, 26, 29, 25, 24
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also number of odd entries in n-th row of triangle of Stirling numbers of the second kind (A008277). - Benoit Cloitre, Feb 28 2004
Apparently (except for the first term) the number of odd entries in the alternated diagonals of Pascal's triangle at 45 degrees slope. - Javier Torres (adaycalledzero(AT)hotmail.com), Jul 26 2009
The Kn3 and Kn4 triangle sums, see A180662 for their definitions, of Sierpiński's triangle A047999 equal a(n+1). - Johannes W. Meijer, Jun 05 2011
From Yosu Yurramendi, Jun 23 2014: (Start)
If the terms (n>1) are written as an array:
2,
3, 3,
4, 5, 5, 4,
5, 7, 8, 7, 7, 8, 7, 5,
6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 6,
7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 11, 16, 19,17,18,
then the sum of the k-th row is 2*3^(k-2), each column is an arithmetic progression. The differences of the arithmetic progressions give the sequence itself (a(2^(m+1)+1+k) - a(2^m+1+k) = a(k+1), m >= 1, 1 <= k <= 2^m), because a(n) = A002487(2*n-1) and A002487 has these properties. A071585 also has these properties. Each row is a palindrome: a(2^(m+1)+1-k) = a(2^m+k), m >= 0, 1 <= k <= 2^m.
If the terms (n>0) are written in this way:
1,
2, 3,
3, 4, 5, 5,
4, 5, 7, 8, 7, 7, 8, 7,
5, 6, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9,
6, 7, 11, 14, 13, 15, 18, 17, 13, 14, 19, 21, 18, 17, 19, 16, 11, 11, 16, 19,
each column is an arithmetic progression and the steps also give the sequence itself (a(2^(m+1)+k) - a(2^m+k) = a(k), m >= 0, 0 <= k < 2^m). Moreover, by removing the first term of each column:
a(2^(m+1)+k) = A049448(2^m+k+1), m >= 0, 0 <= k < 2^m.
(End)
k > 1 occurs in this sequence phi(k) = A000010(k) times. - Franklin T. Adams-Watters, May 25 2015
Except for the initial 1, this is the odd bisection of A002487. The even bisection of A002487 is A002487 itself. - Franklin T. Adams-Watters, May 25 2015
For all m >= 0, max_{k=1..2^m} a(2^m+k) = A000045(m+3) (Fibonacci sequence). - Yosu Yurramendi, Jun 05 2016
For all n >= 2, max(m: a(2^m+k) = n, 1<=k<=2^m) = n-2. - Yosu Yurramendi, Jun 05 2016
a(2^m+1) = m+2, m >= 0; a(2^m+2) = 2m+1, m>=1; min_{m>=0, k=1..2^m} a(2^m+k) = m+2; min_{m>=2, k=2..2^m-1} a(2^m+k) = 2m+1. - Yosu Yurramendi, Jun 06 2016
a(2^(m+2) + 2^(m+1) - k) - a(2^(m+1) + 2^m-k) = 2*a(k+1), m >= 0, 0 <= k <= 2^m. - Yosu Yurramendi, Jun 09 2016
If the initial 1 is omitted, this is the number of nonzero entries in row n of the generalized Pascal triangle P_2, see A282714 [Leroy et al., 2017]. - N. J. A. Sloane, Mar 02 2017
Apparently, this sequence was introduced by Johann Gustav Hermes in 1894. His paper gives a strong connection between this sequence and the so-called "Gaussian brackets" ("Gauss'schen Klammer"). For an independent discussion about Gaussian brackets, see the relevant MathWorld article and the article by Herzberger (1943). Srinivasan (1958) gave another, more modern, explanation of the connection between this sequence and the Gaussian brackets. (Parenthetically, J. G. Hermes is the mathematician who completed or constructed the regular polygon with 65537 sides.) - Petros Hadjicostas, Sep 18 2019

Examples

			[ 0/1; 1/1; ] 1/2; 1/3, 2/3; 1/4, 2/5, 3/5, 3/4; 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5; ...

References

  • P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 61.
  • 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. 158.
  • J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

For numerators see A007305.
Cf. A001222, A002487, A006842, A006843, A047679, A054424, A065674-A065675, A065810, A260443, A277324, A277328, A283986, A283987, A283988, A284009, A284265, A284266, A284267, A284268, A284565, A284566, A285106, A285107, A285108, A287731, A287732.
See also A282714.

Programs

  • Magma
    [1] cat [&+[Binomial(n+k,2*k) mod 2: k in [0..n]]: n in [0..80]]; // Vincenzo Librandi, Jun 10 2019
  • Maple
    A007306 := proc(n): if n=0 then 1 else A002487(2*n-1) fi: end: A002487 := proc(m) option remember: local a, b, n; a := 1; b := 0; n := m; while n>0 do if type(n, odd) then b := a + b else a := a + b end if; n := floor(n/2); end do; b; end proc: seq(A007306(n),n=0..77); # Johannes W. Meijer, Jun 05 2011
  • Mathematica
    a[0] = 1; a[n_] := Sum[ Mod[ Binomial[n+k-1, 2k] , 2], {k, 0, n}]; Table[a[n], {n, 0, 77}] (* Jean-François Alcover, Dec 16 2011, after Paul Barry *)
a[0] = 0; a[1] = 1;
Flatten[{1,Table[a[2*n] = a[n]; a[2*n + 1] = a[n] + a[n + 1], {n, 0, 50}]}] (* Horst H. Manninger, Jun 09 2021 *)
  • PARI
    {a(n) = if( n<1, n==0, n--; sum( k=0, n, binomial( n+k, n-k)%2))};
  • PARI
    {a(n) = my(m); if( n<2, n>=0, m = 2^length( binary( n-1)); a(n - m/2) + a(m-n+1))}; /* Michael Somos, May 30 2005 */
  • Python
    from sympy import binomial
def a(n):
    return 1 if n<1 else sum(binomial(n + k - 1, 2*k) % 2 for k in range(n + 1))
print([a(n) for n in range(101)]) # Indranil Ghosh, Mar 22 2017
  • Python
    from functools import reduce
def A007306(n): return sum(reduce(lambda x,y:(x[0],sum(x)) if int(y) else (sum(x),x[1]),bin((n<<1)-1)[-1:2:-1],(1,0))) if n else 1 # Chai Wah Wu, May 18 2023
  • R
    maxrow <- 6 # by choice
a <- c(1,2)
for(m in 0:maxrow) for(k in 1:2^m){
  a[2^(m+1)+k  ] <- a[2^m+k] + a[k]
  a[2^(m+1)-k+1] <- a[2^m+k]
}
a
# Yosu Yurramendi, Jan 05 2015
  • R
    # Given n, compute directly a(n)
# by taking into account the binary representation of n-1
# aa <- function(n){
  b <- as.numeric(intToBits(n))
  l <- sum(b)
  m <- which(b == 1)-1
  d <- 1
  if(l > 1) for(j in 1:(l-1)) d[j] <- m[j+1]-m[j]+1
  f <- c(1,m[1]+2) # In A002487: f <- c(0,1)
  if(l > 1) for(j in 3:(l+1)) f[j] <- d[j-2]*f[j-1]-f[j-2]
  return(f[l+1])
}
# a(0) = 1, a(1) = 1, a(n) = aa(n-1)   n > 1
#
# Example
n <- 73
aa(n-1)
#
# Yosu Yurramendi, Dec 15 2016
  • Sage
    @CachedFunction
def a(n):
    return a((odd_part(n-1)+1)/2)+a((odd_part(n)+1)/2) if n>1 else 1
[a(n) for n in (0..77)] # after Alessandro De Luca, Peter Luschny, May 20 2014
  • Sage
    def A007306(n):
    if n == 0: return 1
    M = [1, 1]
    for b in (n-1).bits():
        M[b] = M[0] + M[1]
    return M[1]
print([A007306(n) for n in (0..77)]) # Peter Luschny, Nov 28 2017
  • Scheme
    (define (A007306 n) (if (zero? n) 1 (A002487 (+ n n -1)))) ;; Code for A002487 given in that entry. - Antti Karttunen, Mar 21 2017

Formula

Recurrence: a(0) to a(8) are 1, 1, 2, 3, 3, 4, 5, 5, 4; thereafter a(n) = a(n-2^p) + a(2^(p+1)-n+1), where 2^p < n <= 2^(p+1). [J. Hermes, Math. Ann., 1894; quoted by Dickson, Vol. 1, p. 158] - N. J. A. Sloane, Mar 24 2019
a(4*n) = -a(n)+2*a(2*n); a(4*n+1) = -a(n)+a(2*n)+a(2*n+1); a(4*n+2)=a(n)-a(2*n)+2*a(2*n+1); a(4*n+3) = 4*a(n)-4*a(2*n)+3*a(2*n+1). Thus a(n) is a 2-regular sequence. - Jeffrey Shallit, Dec 26 2024
For n > 0, a(n) = A002487(n-1) + A002487(n) = A002487(2*n-1).
a(0) = 1; a(n) = Sum_{k=0..n-1} C(n-1+k, n-1-k) mod 2, n > 0. - Benoit Cloitre, Jun 20 2003
a(n+1) = Sum_{k=0..n} binomial(2*n-k, k) mod 2; a(n) = 0^n + Sum_{k=0..n-1} binomial(2(n-1)-k, k) mod 2. - Paul Barry, Dec 11 2004
a(n) = Sum_{k=0..n} C(n+k,2*k) mod 2. - Paul Barry, Jun 12 2006
a(0) = a(1) = 1; a(n) = a(A003602(n-1)) + a(A003602(n)), n > 1. - Alessandro De Luca, May 08 2014
a(n) = A007305(n+(2^m-1)), m=A029837(n), n=1,2,3,... . - Yosu Yurramendi, Jul 04 2014
a(n) = A007305(2^(m+1)-n) - A007305(2^m-n), m >= (A029837(n)+1), n=1,2,3,... - Yosu Yurramendi, Jul 05 2014
a(2^m) = m+1, a(2^m+1) = m+2 for m >= 0. - Yosu Yurramendi, Jan 01 2015
a(n+2) = A007305(n+2) + A047679(n) n >= 0. - Yosu Yurramendi, Mar 30 2016
a(2^m+2^r+k) = a(2^r+k)(m-r+1) - a(k), m >= 2, 0 <= r <= m-1, 0 <= k < 2^r. Example: a(73) = a(2^6+2^3+1) = a(2^3+1)*(6-3+1) - a(1) = 5*4 - 1 = 19 . - Yosu Yurramendi, Jul 19 2016
From Antti Karttunen, Mar 21 2017 & Apr 12 2017: (Start)
For n > 0, a(n) = A001222(A277324(n-1)) = A001222(A260443(n-1)*A260443(n)).
The following decompositions hold for all n > 0:
a(n) = A277328(n-1) + A284009(n-1).
a(n) = A283986(n) + A283988(n) = A283987(n) + 2*A283988(n).
a(n) = 2*A284265(n-1) + A284266(n-1).
a(n) = A284267(n-1) + A284268(n-1).
a(n) = A284565(n-1) + A284566(n-1).
a(n) = A285106(n-1) + A285108(n-1) = A285107(n-1) + 2*A285108(n-1). (End)
a(A059893(n)) = a(n+1) for n > 0. - Yosu Yurramendi, May 30 2017
a(n) = A287731(n) + A287732(n) for n > 0. - I. V. Serov, Jun 09 2017
a(n) = A287896(n) + A288002(n) for n > 1.
a(n) = A287896(n-1) + A002487(n-1) - A288002(n-1) for n > 1.
a(n) = a(n-1) + A002487(n-1) - 2*A288002(n-1) for n > 1. - I. V. Serov, Jun 14 2017
From Yosu Yurramendi, May 14 2019: (Start)
For m >= 0, M >= m, 0 <= k < 2^m,
a((2^(m+1) + A119608(2^m+k+1))*2^(M-m) - A000035(2^m+k)) =
a((2^(m+2) - A119608(2^m+k+1))*2^(M-m) - A000035(2^m+k)-1) =
a(2^(M+2) - (2^m+k)) = a(2^(M+1) + (2^m+k) + 1) =
a(2^m+k+1)*(M-m) + a(2^(m+1)+2^m+k+1). (End)
a(k) = sqrt(A007305(2^(m+1)+k)*A047679(2^(m+1)+k-2) - A007305(2^m+k)*A047679(2^m+k-2)), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Jun 09 2019
G.f.: 1 + x * (1 + x) * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Jul 19 2019
Conjecture: a(n) = a(n-1) + b(n-1) - 2*(a(n-1) mod b(n-1)) for n > 1 with a(0) = a(1) = 1 where b(n) = a(n) - b(n-1) for n > 1 with b(1) = 1. - Mikhail Kurkov, Mar 13 2022

Extensions

Formula fixed and extended by Franklin T. Adams-Watters, Jul 07 2009
Incorrect Maple program removed by Johannes W. Meijer, Jun 05 2011

A023758 Numbers of the form 2^i - 2^j with i >= j.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 56, 60, 62, 63, 64, 96, 112, 120, 124, 126, 127, 128, 192, 224, 240, 248, 252, 254, 255, 256, 384, 448, 480, 496, 504, 508, 510, 511, 512, 768, 896, 960, 992, 1008, 1016, 1020, 1022, 1023
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Numbers whose digits in base 2 are in nonincreasing order.
Might be called "nialpdromes".
Subset of A077436. Proof: Since a(n) is of the form (2^i-1)*2^j, i,j >= 0, a(n)^2 = (2^(2i) - 2^(i+1))*2^(2j) + 2^(2j) where the first sum term has i-1 one bits and its 2j-th bit is zero, while the second sum term switches the 2j-th bit to one, giving i one bits, as in a(n). - Ralf Stephan, Mar 08 2004
Numbers whose binary representation contains no "01". - Benoit Cloitre, May 23 2004
Every polynomial with coefficients equal to 1 for the leading terms and 0 after that, evaluated at 2. For instance a(13) = x^4 + x^3 + x^2 at 2, a(14) = x^4 + x^3 + x^2 + x at 2. - Ben Paul Thurston, Jan 11 2008
From Gary W. Adamson, Jul 18 2008: (Start)
As a triangle by rows starting:
1;
2, 3;
4, 6, 7;
8, 12, 14, 15;
16, 24, 28, 30, 31;
...,
equals A000012 * A130123 * A000012, where A130123 = (1, 0,2; 0,0,4; 0,0,0,8; ...). Row sums of this triangle = A000337 starting (1, 5, 17, 49, 129, ...). (End)
First differences are A057728 = 1; 1; 1; 1; 2,1; 1; 4,2,1; 1; 8,4,2,1; 1; ... i.e., decreasing powers of 2, separated by another "1". - M. F. Hasler, May 06 2009
Apart from first term, numbers that are powers of 2 or the sum of some consecutive powers of 2. - Omar E. Pol, Feb 14 2013
From Andres Cicuttin, Apr 29 2016: (Start)
Numbers that can be digitally generated with twisted ring (Johnson) counters. This is, the binary digits of a(n) correspond to those stored in a shift register where the input bit of the first bit storage element is the inverted output of the last storage element. After starting with all 0’s, each new state is obtained by rotating the stored bits but inverting at each state transition the last bit that goes to the first position (see link).
Examples: for a(n) represented by three bits
Binary
a(5)= 4 -> 100 last bit = 0
a(6)= 6 -> 110 first bit = 1 (inverted last bit of previous number)
a(7)= 7 -> 111
and for a(n) represented by four bits
Binary
a(8) = 8 -> 1000
a(9) = 12 -> 1100 last bit = 0
a(10)= 14 -> 1110 first bit = 1 (inverted last bit of previous number)
a(11)= 15 -> 1111
(End)
Powers of 2 represented in bases which are terms of this sequence must always contain at least one digit which is also a power of 2. This is because 2^i mod (2^i - 2^j) = 2^j, which means the last digit always cycles through powers of 2 (or if i=j+1 then the first digit is a power of 2 and the rest are trailing zeros). The only known non-member of this sequence with this property is 5. - Ely Golden, Sep 05 2017
Numbers k such that k = 2^(1 + A000523(k)) - 2^A007814(k). - Daniel Starodubtsev, Aug 05 2021
A002260(n) = v(a(n)/2^v(a(n))+1) and A002024(n) = A002260(n) + v(a(n)) where v is the dyadic valuation (i.e., A007814). - Lorenzo Sauras Altuzarra, Feb 01 2023

Examples

			a(22) = 64 = 32 + 32 = 2^5 + a(16) = 2^A003056(20) + a(22-5-1).
a(23) = 96 = 64 + 32 = 2^6 + a(16) = 2^A003056(21) + a(23-6-1).
a(24) = 112 = 64 + 48 = 2^6 + a(17) = 2^A003056(22) + a(24-6-1).

Links

Crossrefs

A000337(r) = sum of row T(r, c) with 0 <= c < r. See also A002024, A003056, A140129, A140130, A221975.
Cf. A007088, A130123, A101082 (complement), A340375 (characteristic function).
This is the base-2 version of A064222. First differences are A057728.
Subsequence of A077436, of A129523, of A277704, and of A333762.
Subsequences: A043569 (nonzero even terms, or equally, nonzero terms doubled), A175332, A272615, A335431, A000396 (its even terms only), A324200.
Positions of zeros in A049502, A265397, A277899, A284264.
Positions of ones in A283983, A283989.
Positions of nonzero terms in A341509 (apart from the initial zero).
Positions of squarefree terms in A260443.
Fixed points of A264977, A277711, A283165, A334666.
Distinct terms in A340632.
Cf. also A002024, A002260, A007814, A133797, A152449, A181666, A211705, A277699, A306441.
Cf. also A309758, A309759, A309761 (for analogous sequences).

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a023758 n = a023758_list !! (n-1)
a023758_list = 0 : f (singleton 1) where
f s = x : f (if even x then insert z s' else insert z $ insert (z+1) s')
where z = 2*x; (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 24 2014, Dec 19 2012
  • Maple
    a:=proc(n) local n2,d: n2:=convert(n,base,2): d:={seq(n2[j]-n2[j-1],j=2..nops(n2))}: if n=0 then 0 elif n=1 then 1 elif d={0,1} or d={0} or d={1} then n else fi end: seq(a(n),n=0..2100); # Emeric Deutsch, Apr 22 2006
  • Mathematica
    Union[Flatten[Table[2^i - 2^j, {i, 0, 100}, {j, 0, i}]]] (* T. D. Noe, Mar 15 2011 *)
Select[Range[0, 2^10], NoneTrue[Differences@ IntegerDigits[#, 2], # > 0 &] &] (* Michael De Vlieger, Sep 05 2017 *)
  • PARI
    for(n=0,2500,if(prod(k=1,length(binary(n))-1,component(binary(n),k)+1-component(binary(n),k+1))>0,print1(n,",")))
  • PARI
    A023758(n)= my(r=round(sqrt(2*n--))); (1<<(n-r*(r-1)/2)-1)<<(r*(r+1)/2-n)
/* or, to illustrate the "decreasing digit" property and analogy to A064222: */
A023758(n,show=0)={ my(a=0); while(n--, show & print1(a","); a=vecsort(binary(a+1)); a*=vector(#a,j,2^(j-1))~); a} \\ M. F. Hasler, May 06 2009
  • PARI
    is(n)=if(n<5,1,n>>=valuation(n,2);n++;n>>valuation(n,2)==1) \\ Charles R Greathouse IV, Jan 04 2016
  • PARI
    list(lim)=my(v=List([0]),t); for(i=1,logint(lim\1+1,2), t=2^i-1; while(t<=lim, listput(v,t); t*=2)); Set(v) \\ Charles R Greathouse IV, May 03 2016
  • Python
    def a_next(a_n): return (a_n | (a_n >> 1)) + (a_n & 1)
a_n = 1; a = [0]
for i in range(55): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022
  • Python
    from math import isqrt
def A023758(n): return (1<<(m:=isqrt(n-1<<3)+1>>1))-(1<<(m*(m+1)-(n-1<<1)>>1)) # Chai Wah Wu, Feb 23 2025

Formula

a(n) = 2^s(n) - 2^((s(n)^2 + s(n) - 2n)/2) where s(n) = ceiling((-1 + sqrt(1+8n))/2). - Sam Alexander, Jan 08 2005
a(n) = 2^k + a(n-k-1) for 1 < n and k = A003056(n-2). The rows of T(r, c) = 2^r-2^c for 0 <= c < r read from right to left produce this sequence: 1; 2, 3; 4, 6, 7; 8, 12, 14, 15; ... - Frank Ellermann, Dec 06 2001
For n > 0, a(n) mod 2 = A010054(n). - Benoit Cloitre, May 23 2004
A140130(a(n)) = 1 and for n > 1: A140129(a(n)) = A002262(n-2). - Reinhard Zumkeller, May 14 2008
a(n+1) = (2^(n - r(r-1)/2) - 1) 2^(r(r+1)/2 - n), where r=round(sqrt(2n)). - M. F. Hasler, May 06 2009
Start with A000225. If k is in the sequence, then so is 2k. - Ralf Stephan, Aug 16 2013
G.f.: (x^2/((2-x)*(1-x)))*(1 + Sum_{k>=0} x^((k^2+k)/2)*(1 + x*(2^k-1))). The sum is related to Jacobi theta functions. - Robert Israel, Feb 24 2015
A049502(a(n)) = 0. - Reinhard Zumkeller, Jun 17 2015
a(n) = a(n-1) + a(n-d)/a(d*(d+1)/2 + 2) if n > 1, d > 0, where d = A002262(n-2). - Yuchun Ji, May 11 2020
A277699(a(n)) = a(n)^2, A306441(a(n)) = a(n+1). - Antti Karttunen, Feb 15 2021 (the latter identity from A306441)
Sum_{n>=2} 1/a(n) = A211705. - Amiram Eldar, Feb 20 2022

Extensions

Definition changed by N. J. A. Sloane, Jan 05 2008

A125184 Triangle read by rows: T(n,k) is the coefficient of t^k in the Stern polynomial B(n,t) (n>=0, k>=0).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 2, 1, 3, 1, 0, 0, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 3, 3, 0, 0, 1, 2, 1, 4, 3, 0, 1, 3, 1, 1, 3, 2, 1, 0, 0, 0, 1, 1, 1, 2, 3, 1, 0, 1, 2, 2, 1, 3, 3, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 0, 1, 1, 1
Offset: 0

Views

Author

Emeric Deutsch, Dec 04 2006

Keywords

Comments

The Stern polynomials B(n,t) are defined by B(0,t)=0, B(1,t)=1, B(2n,t)=tB(n,t), B(2n+1,t)=B(n+1,t)+B(n,t) for n>=1 (see S. Klavzar et al.).
Also number of hyperbinary representations of n-1 containing exactly k digits 1. A hyperbinary representation of a nonnegative integer n is a representation of n as a sum of powers of 2, each power being used at most twice. Example: row 9 of the triangle is 1,2,1; indeed the hyperbinary representations of 8 are 200 (2*2^2+0*2^1+0*2^0), 120 (1*2^2+2*2^1+0*2^0), 1000 (1*2^3+0*2^2+0*2^1+0*2^0) and 112 (1*2^2+1*2^1+2*1^0), having 0,1,1 and 2 digits 1, respectively (see S. Klavzar et al. Corollary 3).
Number of terms in row n is A277329(n) (= 1+A057526(n) for n >= 1).
Row sums yield A002487 (Stern's diatomic series).
T(2n+1,1) = A005811(n) = number of 1's in the standard Gray code of n (S. Klavzar et al. Theorem 8). T(4n+1,1)=1, T(4n+3,1)=0 (S. Klavzar et al., Lemma 5).
From Antti Karttunen, Oct 27 2016: (Start)
Number of nonzero terms on row n is A277314(n).
Number of odd terms on row n is A277700(n).
Maximal term on row n is A277315(n).
Product of nonzero terms on row n is A277325(n).
Number of times where row n and n+1 both contain nonzero term in the same position is A277327(n).
(End)

Examples

			Triangle starts:
0;
1;
0, 1;
1, 1;
0, 0, 1;
1, 2;
0, 1, 1;
1, 1, 1;
0, 0, 0, 1;
1, 2, 1;
0, 1, 2;
1, 3, 1;

Links

Crossrefs

Cf. A057526, A002487, A005811.
Cf. A186890 (n such that the Stern polynomial B(n,x) is self-reciprocal).
Cf. A186891 (n such that the Stern polynomial B(n,x) is irreducible).
Cf. A260443 (Stern polynomials encoded in the prime factorization of n).
Cf. also A277314, A277315, A277325, A277327, A277329, A277700.

Programs

  • Maple
    B:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then t*B(n/2) else B((n+1)/2)+B((n-1)/2) fi end: for n from 0 to 36 do B(n):=sort(expand(B(n))) od: dg:=n->degree(B(n)): 0; for n from 0 to 40 do seq(coeff(B(n),t,k),k=0..dg(n)) od; # yields sequence in triangular form
  • Mathematica
    B[0, ] = 0; B[1, ] = 1; B[n_, t_] := B[n, t] = If[EvenQ[n], t*B[n/2, t], B[1 + (n-1)/2, t] + B[(n-1)/2, t]]; row[n_] := CoefficientList[B[n, t], t]; row[0] = {0}; Array[row, 40, 0] // Flatten (* Jean-François Alcover, Jul 30 2015 *)

Extensions

0 prepended by T. D. Noe, Feb 28 2011
Original comment slightly edited by Antti Karttunen, Oct 27 2016

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600
Offset: 0

Views

Author

Antti Karttunen, Mar 16 2017

Keywords

Comments

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.
This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:
1
|
...................2....................
6 12
30......../ \........60 180......../ \......360
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
210 420 1260 2520 6300 12600 37800 75600
etc.

Links

Crossrefs

Cf. A129912 (same terms, but sorted into ascending order).
Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005, A324287, A324289, A324341, A324342, A324343.
Cf. A005940, A052330, A322827, A323505 for other similar trees.
Cf. also A260443.

Programs

  • Mathematica
    Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)
  • PARI
    A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019
  • Python
    from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a108951(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017
  • Python
    from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022
  • Scheme
    (define (A283477 n) (A108951 (A019565 n)))
;; Recursive "binary tree" implementation, using memoization-macro definec:
(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

Formula

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
a(n) = A108951(A019565(n)).
A097248(a(n)) = A283475(n).
A007814(a(n)) = A051903(a(n)) = A000120(n).
A001221(a(n)) = A070939(n).
A001222(a(n)) = A029931(n).
A048675(a(n)) = A005187(n).
A248663(a(n)) = A006068(n).
A090880(a(n)) = A283483(n).
A276075(a(n)) = A283984(n).
A276085(a(n)) = A283985(n).
A046660(a(n)) = A124757(n).
A056169(a(n)) = A065120(n). [seems to be]
A005361(a(n)) = A284001(n).
A072411(a(n)) = A284002(n).
A007913(a(n)) = A284003(n).
A000005(a(n)) = A284005(n).
A324286(a(n)) = A324287(n).
A276086(a(n)) = A324289(n).
A267263(a(n)) = A324341(n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

Extensions

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017
Four more linking formulas added by Antti Karttunen, Feb 25 2019

A072411 LCM of exponents in prime factorization of n, a(1) = 1.

Original entry on oeis.org

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

Views

Author

Labos Elemer, Jun 17 2002

Keywords

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 14, 168, 1779, 17959, 180665, 1808044, 18084622, 180856637, 1808585068, 18085891506, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.8085... . - Amiram Eldar, Sep 10 2022

Examples

			n = 288 = 2*2*2*2*2*3*3; lcm(5,2) = 10; Product(5,2) = 10, max(5,2) = 5;
n = 180 = 2*2*3*3*5; lcm(2,2,1) = 2; Product(2,2,1) = 4; max(2,2,1) = 2; it deviates both from maximum of exponents (A051903, for the first time at n=72), and product of exponents (A005361, for the first time at n=36).
For n = 36 = 2*2*3*3 = 2^2 * 3^2 we have a(36) = lcm(2,2) = 2.
For n = 72 = 2*2*2*3*3 = 2^3 * 3^2 we have a(72) = lcm(2,3) = 6.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(2,4) = 4.
For n = 360 = 2^3 * 3^2 * 5^1 we have a(360) = lcm(1,2,3) = 6.

Links

Crossrefs

Cf. A028234, A067029, A072412-A072414, A273058, A284569, A290103.
Similar sequences: A001222 (sum of exponents), A005361 (product), A051903 (maximal exponent), A051904 (minimal exponent), A052409 (gcd of exponents), A267115 (bitwise-and), A267116 (bitwise-or), A268387 (bitwise-xor).
Cf. also A055092, A060131.
Differs from A290107 for the first time at n=144.
After the initial term, differs from A157754 for the first time at n=360.

Programs

  • Mathematica
    Table[LCM @@ Last /@ FactorInteger[n], {n, 2, 100}] (* Ray Chandler, Jan 24 2006 *)
  • PARI
    a(n) = lcm(factor(n)[,2]); \\ Michel Marcus, Mar 25 2017
  • Python
    from sympy import lcm, factorint
def a(n):
    l=[]
    f=factorint(n)
    for i in f: l+=[f[i],]
    return lcm(l)
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 25 2017

Formula

a(1) = 1; for n > 1, a(n) = lcm(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 09 2016
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A284569(A156552(n)).
a(n) = A290103(A181819(n)).
a(A289625(n)) = A002322(n).
a(A290095(n)) = A055092(n).
a(A275725(n)) = A060131(n).
a(A260443(n)) = A277326(n).
a(A283477(n)) = A284002(n). (End)

Extensions

a(1) = 1 prepended and the data section filled up to 120 terms by Antti Karttunen, Aug 09 2016

A248663 Binary encoding of the prime factors of the squarefree part of n.

Original entry on oeis.org

0, 1, 2, 0, 4, 3, 8, 1, 0, 5, 16, 2, 32, 9, 6, 0, 64, 1, 128, 4, 10, 17, 256, 3, 0, 33, 2, 8, 512, 7, 1024, 1, 18, 65, 12, 0, 2048, 129, 34, 5, 4096, 11, 8192, 16, 4, 257, 16384, 2, 0, 1, 66, 32, 32768, 3, 20, 9, 130, 513, 65536, 6, 131072, 1025, 8, 0, 36, 19
Offset: 1

Views

Author

Peter Kagey, Jan 11 2015

Keywords

Comments

The binary digits of a(n) encode the prime factorization of A007913(n), where the i-th digit from the right is 1 if and only if prime(i) divides A007913(n), otherwise 0. - Robert Israel, Jan 12 2015
Old name: a(1) = 0; a(A000040(n)) = 2^(n-1), and a(n*m) = a(n) XOR a(m).
XOR is the bitwise exclusive or operation (A003987).
a(k^2) = 0 for a natural number k.
Equivalently, the i-th binary digit from the right is 1 iff prime(i) divides n an odd number of times, otherwise zero. - Ethan Beihl, Oct 15 2016
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443, with scheme explained in A206284), then A048675(n) gives the evaluation of that polynomial at x=2. This sequence is otherwise similar, except the polynomial is evaluated over the field GF(2), which implies also that all its coefficients are essentially reduced modulo 2. - Antti Karttunen, Dec 11 2015
Squarefree numbers (A005117) give the positions k where a(k) = A048675(k). - Antti Karttunen, Oct 29 2016
From Peter Munn, Jun 07 2021: (Start)
When we encode polynomials with nonnegative integer coefficients as described by Antti Karttunen above, polynomial addition is represented by integer multiplication, multiplication is represented by A297845(.,.), and this sequence represents a surjective semiring homomorphism to polynomials in GF(2)[x] (encoded as described in A048720). The mapping of addition operations by this homomorphism is part of the sequence definition: "a(n*m) = a(n) XOR a(m)". The mapping of multiplication is given by a(A297845(n, k)) = A048720(a(n), a(k)).
In a related way, A329329 defines a representation of a different set of polynomials as positive integers, namely polynomials in GF(2)[x,y].
Let P_n(x,y) denote the polynomial represented, as in A329329, by n >= 1. If 0 is substituted for y in P_n(x,y), we get a polynomial P'_n(x,y) (in which y does not appear, of course) that is equivalent to a polynomial P'_n(x) in GF(2)[x]. a(n) is the integer encoding of P'_n(x) (described in A048720).
Viewed as above, this sequence represents another surjective homomorphism, a homomorphism between polynomial rings, with A329329(.,.)/A059897(.,.) and A048720(.,.)/A003987(.,.) as the respective ring operations.
a(n) can be composed as a(n) = A048675(A007913(n)) and the effect of the A007913(.) component corresponds to different operations on the respective polynomial domains of the two homomorphisms described above. In the first homomorphism, coefficients are reduced modulo 2; in the second, 0 is substituted for y. This is illustrated in the examples.
(End)

Examples

			a(3500) = a(2^2 * 5^3 * 7) = a(2) XOR a(2) XOR a(5) XOR a(5) XOR a(5) XOR a(7) = 1 XOR 1 XOR 4 XOR 4 XOR 4 XOR 8 = 0b0100 XOR 0b1000 = 0b1100 = 12.
From _Peter Munn_, Jun 07 2021: (Start)
The examples in the table below illustrate the homomorphisms (between polynomial structures) represented by this sequence.
The staggering of the rows is to show how the mapping n -> A007913(n) -> A048675(A007913(n)) = a(n) relates to the encoded polynomials, as not all encodings are relevant at each stage.
For an explanation of each polynomial encoding, see the sequence referenced in the relevant column heading. (Note also that A007913 generates squarefree numbers, and with these encodings, all squarefree numbers represent equivalent polynomials in N[x] and GF(2)[x,y].)
                     |<-----    encoded polynomials    ----->|
  n  A007913(n) a(n) |         N[x]    GF(2)[x,y]    GF(2)[x]|
                     |Cf.:  A206284       A329329     A048720|
--------------------------------------------------------------
  24                            x+3         x+y+1
          6                     x+1           x+1
                  3                                       x+1
--------------------------------------------------------------
  36                           2x+2          xy+y
          1                       0             0
                  0                                         0
--------------------------------------------------------------
  60                        x^2+x+2       x^2+x+y
         15                   x^2+x         x^2+x
                  6                                     x^2+x
--------------------------------------------------------------
  90                       x^2+2x+1      x^2+xy+1
         10                   x^2+1         x^2+1
                  5                                     x^2+1
--------------------------------------------------------------
This sequence is a left inverse of A019565. A019565(.) maps a(n) to A007913(n) for all n, effectively reversing the second stage of the mapping from n to a(n) shown above. So, with the encodings used here, A019565(.) represents each of two injective homomorphisms that map polynomials in GF(2)[x] to equivalent polynomials in N[x] and GF(2)[x,y] respectively.
(End)

Links

Crossrefs

Cf. A000040, A000079, A000720, A005117, A020639, A032742, A048672, A055396, A100112, A206284.
A048675 composed with A007913. A007814 composed with A225546.
A homomorphism from A297845/A003991 and from A329329/A059897 to A048720/A003987, mapping A206296 to A168081, A260443 to A264977, A265408 to A265407, A275734 to A275808, A276076 to A276074, A283477 to A006068.
A left inverse of A019565.
Other sequences used to express relationship between terms of this sequence: A003961, A007913, A331590, A334747.
Cf. also A099884, A277330.
A087207 is the analogous sequence with OR.
A277417 gives the positions where coincides with A277333.
A000290 gives the positions of zeros.

Programs

  • Haskell
    import Data.Bits (xor)
a248663 = foldr (xor) 0 . map (\i -> 2^(i - 1)) . a112798_row
-- Peter Kagey, Sep 16 2016
  • Maple
    f:= proc(n)
local F,f;
F:= select(t -> t[2]::odd, ifactors(n)[2]);
add(2^(numtheory:-pi(f[1])-1), f = F)
end proc:
seq(f(i),i=1..100); # Robert Israel, Jan 12 2015
  • Mathematica
    a[1] = 0; a[n_] := a[n] = If[PrimeQ@ n, 2^(PrimePi@ n - 1), BitXor[a[#], a[n/#]] &@ FactorInteger[n][[1, 1]]]; Array[a, 66] (* Michael De Vlieger, Sep 16 2016 *)
  • PARI
    A248663(n) = vecsum(apply(p -> 2^(primepi(p)-1),factor(core(n))[,1])); \\ Antti Karttunen, Feb 15 2021
  • Python
    from sympy import factorint, primepi
from sympy.ntheory.factor_ import core
def a048675(n):
    f=factorint(n)
    return 0 if n==1 else sum([f[i]*2**(primepi(i) - 1) for i in f])
def a(n): return a048675(core(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 21 2017
  • Ruby
    require 'prime'
def f(n)
  a = 0
  reverse_primes = Prime.each(n).to_a.reverse
  reverse_primes.each do |prime|
    a <<= 1
    while n % prime == 0
      n /= prime
      a ^= 1
    end
  end
  a
end
(Scheme, with memoizing-macro definec)
(definec (A248663 n) (cond ((= 1 n) 0) ((= 1 (A010051 n)) (A000079 (- (A000720 n) 1))) (else (A003987bi (A248663 (A020639 n)) (A248663 (A032742 n)))))) ;; Where A003987bi computes bitwise-XOR as in A003987.
;; Alternatively:
(definec (A248663 n) (cond ((= 1 n) 0) (else (A003987bi (A000079 (- (A055396 n) 1)) (A248663 (A032742 n))))))
;; Antti Karttunen, Dec 11 2015

Formula

a(1) = 0; for n > 1, if n is a prime, a(n) = 2^(A000720(n)-1), otherwise a(A020639(n)) XOR a(A032742(n)). [After the definition.] - Antti Karttunen, Dec 11 2015
For n > 1, this simplifies to: a(n) = 2^(A055396(n)-1) XOR a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n. Cf. a similar formula for A048675.]
Other identities and observations. For all n >= 0:
a(n) = A048672(A100112(A007913(n))). - Peter Kagey, Dec 10 2015
From Antti Karttunen, Dec 11 2015, Sep 19 & Oct 27 2016, Feb 15 2021: (Start)
a(n) = a(A007913(n)). [The result depends only on the squarefree part of n.]
a(n) = A048675(A007913(n)).
a(A206296(n)) = A168081(n).
a(A260443(n)) = A264977(n).
a(A265408(n)) = A265407(n).
a(A275734(n)) = A275808(n).
a(A276076(n)) = A276074(n).
a(A283477(n)) = A006068(n).
(End)
From Peter Munn, Jan 09 2021 and Apr 20 2021: (Start)
a(n) = A007814(A225546(n)).
a(A019565(n)) = n; A019565(a(n)) = A007913(n).
a(A003961(n)) = 2 * a(n).
a(A297845(n, k)) = A048720(a(n), a(k)).
a(A329329(n, k)) = A048720(a(n), a(k)).
a(A059897(n, k)) = A003987(a(n), a(k)).
a(A331590(n, k)) = a(n) + a(k).
a(A334747(n)) = a(n) + 1.
(End)

Extensions

New name from Peter Munn, Nov 01 2023

A186891 Numbers n such that the Stern polynomial B(n,x) is irreducible.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 157, 161, 163, 167, 169, 173, 175, 179, 181, 185, 191, 193, 197, 199
Offset: 1

Views

Author

T. D. Noe, Feb 28 2011

Keywords

Comments

Ulas and Ulas conjecture that all primes are here. The nonprime n are in A186892. See A186886 for the least number having n prime factors.

Links

Crossrefs

Cf. A057526 (degree of Stern polynomials), A125184, A260443 (Stern polynomials).
Cf. A186892 (subsequence of nonprime terms).
Cf. A186893 (subsequence for self-reciprocal polynomials).
Positions of 0 and 1's in A277013, Positions of 1 and 2's in A284011.
Cf. A283991 (characteristic function for terms > 1).
Cf. also A186886, A277190, A277318, A283992.

Programs

  • Mathematica
    ps[n_] := ps[n] = If[n<2, n, If[OddQ[n], ps[Quotient[n, 2]] + ps[Quotient[n, 2] + 1], x ps[Quotient[n, 2]]]];
selQ[n_] := IrreduciblePolynomialQ[ps[n]];
Join[{1}, Select[Range[200], selQ]] (* Jean-François Alcover, Nov 02 2018, translated from PARI *)
  • PARI
    ps(n)=if(n<2, n, if(n%2, ps(n\2)+ps(n\2+1), 'x*ps(n\2)))
is(n)=polisirreducible(ps(n)) \\ Charles R Greathouse IV, Apr 07 2015

Formula

From Antti Karttunen, Mar 21 2017: (Start)
A283992(a(1+n)) = n.
A260443(a(1+n)) = A277318(n).
(End)

A276075 a(1) = 0, a(n) = (e1*i1! + e2*i2! + ... + ez*iz!) for n = prime(i1)^e1 * prime(i2)^e2 * ... * prime(iz)^ez, where prime(k) is the k-th prime, A000040(k).

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 24, 3, 4, 7, 120, 4, 720, 25, 8, 4, 5040, 5, 40320, 8, 26, 121, 362880, 5, 12, 721, 6, 26, 3628800, 9, 39916800, 5, 122, 5041, 30, 6, 479001600, 40321, 722, 9, 6227020800, 27, 87178291200, 122, 10, 362881, 1307674368000, 6, 48, 13, 5042, 722, 20922789888000, 7, 126, 27, 40322, 3628801, 355687428096000, 10, 6402373705728000, 39916801, 28, 6, 726, 123
Offset: 1

Views

Author

Antti Karttunen, Aug 18 2016

Keywords

Comments

Additive with a(p^e) = e * (PrimePi(p)!), where PrimePi(n) = A000720(n).
a(3181) has 1001 decimal digits. - Michael De Vlieger, Dec 24 2017

Links

Crossrefs

Cf. A000040, A000142, A000720, A002110, A007489, A019565, A028234, A033312, A055396, A059590, A067029, A076954, A206296, A260443, A276080, A276081.
Left inverse of A276076.
Cf. also A048675.

Programs

  • Mathematica
    Array[If[# == 1, 0, Total[FactorInteger[#] /. {p_, e_} /; p > 1 :> e PrimePi[p]!]] &, 66] (* Michael De Vlieger, Dec 24 2017 *)
  • Python
    from sympy import factorint, factorial as f, primepi
def a(n):
    F=factorint(n)
    return 0 if n==1 else sum(F[i]*f(primepi(i)) for i in F)
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Jun 21 2017

Formula

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A000142(A055396(n))).
Other identities.
For all n >= 0:
a(A276076(n)) = n.
a(A002110(n)) = A007489(n).
a(A019565(n)) = A059590(n).
a(A206296(n)) = A276080(n).
a(A260443(n)) = A276081(n).
For all n >= 1:
a(A000040(n)) = n! = A000142(n).
a(A076954(n-1)) = A033312(n).

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 5, 13, 14, 15, 5, 17, 10, 19, 15, 21, 22, 23, 10, 7, 26, 15, 21, 29, 30, 31, 10, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 7, 46, 47, 15, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 7, 61, 62, 35, 15, 65, 66, 67, 51, 69, 70, 71, 30, 73, 74, 21
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 03 2004

Keywords

Comments

a(n) = r(n,m) with m such that r(n,m)=r(n,m+1), where r(n,k) = A097246(r(n,k-1)), r(n,0)=n. (The original definition.)
A097248(n) = r(n,a(n)).
From Antti Karttunen, Nov 15 2016: (Start)
The above remark could be interpreted to mean that A097249(n) <= a(n).
All terms are squarefree, and the squarefree numbers are the fixed points.
These are also fixed points eventually reached when iterating A277886.
(End)

Links

Crossrefs

Range of values is A005117.
Cf. A008683, A019565, A048675, A097246, A097247, A097249, A277905, A332824.
A003961, A225546, A277885, A277886, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A007913, A260443, A329050, A329332.
See comments/formulas in A283475, A283478, A331751 giving their relationship to this sequence.

Programs

  • Mathematica
    Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, n], {n, 75}] (* Michael De Vlieger, Mar 18 2017 *)
  • PARI
    A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
\\ Antti Karttunen, Mar 18 2017
  • Python
    from sympy import factorint, nextprime
from operator import mul
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k # Indranil Ghosh, May 15 2017
  • Scheme
    ;; with memoization-macro definec
;; Two implementations:
(definec (A097248 n) (if (not (zero? (A008683 n))) n (A097248 (A097246 n))))
(definec (A097248 n) (if (zero? (A277885 n)) n (A097248 (A277886 n))))
;; Antti Karttunen, Nov 15 2016

Formula

a(A005117(n)) = A005117(n).
From Antti Karttunen, Nov 15 2016: (Start)
If A008683(n) <> 0 [when n is squarefree], a(n) = n, otherwise a(n) = a(A097246(n)).
If A277885(n) = 0, a(n) = n, otherwise a(n) = a(A277886(n)).
A007913(a(n)) = a(n).
a(A007913(n)) = A007913(n).
A048675(a(n)) = A048675(n).
a(A260443(n)) = A019565(n).
(End)
From Peter Munn, Feb 06 2020: (Start)
a(1) = 1; a(p) = p, for prime p; a(m*k) = A331590(a(m), a(k)).
a(A331590(m,k)) = A331590(a(m), a(k)).
a(n^2) = a(A003961(n)) = A003961(a(n)).
a(A225546(n)) = a(n).
a(n) = A225546(2^A048675(n)) = A019565(A048675(n)).
a(A329050(n,k)) = prime(n+k-1) = A000040(n+k-1).
a(A329332(n,k)) = A019565(n * k).
Equivalently, a(A019565(n)^k) = A019565(n * k).
(End)
From Antti Karttunen, Feb 22-25 & Mar 01 2020: (Start)
a(A019565(x)*A019565(y)) = A019565(x+y).
a(A332461(n)) = A332462(n).
a(A332824(n)) = A019565(n).
a(A277905(n,k)) = A277905(n,1) = A019565(n), for all n >= 1, and 1 <= k <= A018819(n).
(End)

Extensions

Name changed and the original definition moved to the Comments section by Antti Karttunen, Nov 15 2016

A264977 a(0) = 0, a(1) = 1, a(2*n) = 2*a(n), a(2*n+1) = a(n) XOR a(n+1).

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 7, 8, 5, 2, 7, 12, 1, 14, 15, 16, 13, 10, 7, 4, 5, 14, 11, 24, 13, 2, 15, 28, 1, 30, 31, 32, 29, 26, 7, 20, 13, 14, 3, 8, 1, 10, 11, 28, 5, 22, 19, 48, 21, 26, 15, 4, 13, 30, 19, 56, 29, 2, 31, 60, 1, 62, 63, 64, 61, 58, 7, 52, 29, 14, 19, 40, 25, 26, 3, 28, 13, 6, 11, 16, 9, 2, 11, 20, 1, 22
Offset: 0

Views

Author

Antti Karttunen, Dec 10 2015

Keywords

Comments

a(n) is the n-th Stern polynomial (cf. A260443, A125184) evaluated at X = 2 over the field GF(2).
For n >= 1, a(n) gives the index of the row where n occurs in array A277710.

Examples

			In this example, binary numbers are given zero-padded to four bits.
a(2) = 2a(1) = 2 * 1 = 2.
a(3) = a(1) XOR a(2) = 1 XOR 2 = 0001 XOR 0010 = 0011 = 3.
a(4) = 2a(2) = 2 * 2 = 4.
a(5) = a(2) XOR a(3) = 2 XOR 3 = 0010 XOR 0011 = 0001 = 1.
a(6) = 2a(3) = 2 * 3 = 6.
a(7) = a(3) XOR a(4) = 3 XOR 4 = 0011 XOR 0100 = 0111 = 7.

Links

Crossrefs

Cf. A000225, A001511, A002487, A003987, A010060, A011655, A048675, A055396, A125184, A248663, A260443, A265397, A277330.
Cf. A023758 (the fixed points).
Cf. also A068156, A168081, A265407.
Cf. A277700 (binary weight of terms).
Cf. A277701, A277712, A277713 (positions of 1's, 2's and 3's in this sequence).
Cf. A277711 (position of the first occurrence of each n in this sequence).
Cf. also arrays A277710, A099884.

Programs

  • Mathematica
    recurXOR[0] = 0; recurXOR[1] = 1; recurXOR[n_] := recurXOR[n] = If[EvenQ[n], 2recurXOR[n/2], BitXor[recurXOR[(n - 1)/2 + 1], recurXOR[(n - 1)/2]]]; Table[recurXOR[n], {n, 0, 100}] (* Jean-François Alcover, Oct 23 2016 *)
  • Python
    class Memoize:
    def _init_(self, func):
        self.func=func
        self.cache={}
    def _call_(self, arg):
        if arg not in self.cache:
            self.cache[arg] = self.func(arg)
        return self.cache[arg]
@Memoize
def a(n): return n if n<2 else 2*a(n//2) if n%2==0 else a((n - 1)//2)^a((n + 1)//2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 27 2017

Formula

a(0) = 0, a(1) = 1, a(2*n) = 2*a(n), a(2*n+1) = a(n) XOR a(n+1).
a(n) = A248663(A260443(n)).
a(n) = A048675(A277330(n)). - Antti Karttunen, Oct 27 2016
Other identities. For all n >= 0:
a(n) = n - A265397(n).
From Antti Karttunen, Oct 28 2016: (Start)
A000035(a(n)) = A000035(n). [Preserves the parity of n.]
A010873(a(n)) = A010873(n). [a(n) mod 4 = n mod 4.]
A001511(a(n)) = A001511(n) = A055396(A277330(n)). [In general, the 2-adic valuation of n is preserved.]
A010060(a(n)) = A011655(n).
a(n) <= n.
For n >= 2, a((2^n)+1) = (2^n) - 3.
The following two identities are so far unproved:
For n >= 2, a(3*A000225(n)) = a(A068156(n)) = 5.
For n >= 2, a(A068156(n)-2) = A062709(n) = 2^n + 3.
(End)
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