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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A002487 Stern's diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also called fusc(n) [Dijkstra].
a(n)/a(n+1) runs through all the reduced nonnegative rationals exactly once [Stern; Calkin and Wilf].
If the terms are written as an array:
column 0 1 2 3 4 5 6 7 8 9 ...
row 0: 0
row 1: 1
row 2: 1,2
row 3: 1,3,2,3
row 4: 1,4,3,5,2,5,3,4
row 5: 1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5
row 6: 1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,...
...
then (ignoring row 0) the sum of the k-th row is 3^(k-1), each column is an arithmetic progression and the steps are nothing but the original sequence. - Takashi Tokita (butaneko(AT)fa2.so-net.ne.jp), Mar 08 2003
From N. J. A. Sloane, Oct 15 2017: (Start)
The above observation can be made more precise. Let A(n,k), n >= 0, 0 <= k <= 2^(n-1)-1 for k > 0, denote the entry in row n and column k of the left-justified array above.
The equations for columns 0,1,2,3,4,... are successively (ignoring row 0):
1 (n >= 1),
n (n >= 2),
n-1 (n >= 3),
2n-3 (n >= 3),
n-2 (n >= 4),
3n-7 (n >= 4),
...
and in general column k > 0 is given by
A(n,k) = a(k)*n - A156140(k) for n >= ceiling(log_2(k+1))+1, and 0 otherwise.
(End)
a(n) is the number of odd Stirling numbers S_2(n+1, 2r+1) [Carlitz].
Moshe Newman proved that the fraction a(n+1)/a(n+2) can be generated from the previous fraction a(n)/a(n+1) = x by 1/(2*floor(x) + 1 - x). The successor function f(x) = 1/(floor(x) + 1 - frac(x)) can also be used.
a(n+1) = number of alternating bit sets in n [Finch].
If f(x) = 1/(1 + floor(x) - frac(x)) then f(a(n-1)/a(n)) = a(n)/a(n+1) for n >= 1. If T(x) = -1/x and f(x) = y, then f(T(y)) = T(x) for x > 0. - Michael Somos, Sep 03 2006
a(n+1) is the number of ways of writing n as a sum of powers of 2, each power being used at most twice (the number of hyperbinary representations of n) [Carlitz; Lind].
a(n+1) is the number of partitions of the n-th integer expressible as the sum of distinct even-subscripted Fibonacci numbers (= A054204(n)), into sums of distinct Fibonacci numbers [Bicknell-Johnson, theorem 2.1].
a(n+1) is the number of odd binomial(n-k, k), 0 <= 2*k <= n. [Carlitz], corrected by Alessandro De Luca, Jun 11 2014
a(2^k) = 1. a(3*2^k) = a(2^(k+1) + 2^k) = 2. Sequences of terms between a(2^k) = 1 and a(2^(k+1)) = 1 are palindromes of length 2^k-1 with a(2^k + 2^(k-1)) = 2 in the middle. a(2^(k-1) + 1) = a(2^k - 1) = k+1 for k > 1. - Alexander Adamchuk, Oct 10 2006
The coefficients of the inverse of the g.f. of this sequence form A073469 and are related to binary partitions A000123. - Philippe Flajolet, Sep 06 2008
It appears that the terms of this sequence are the number of odd entries in the diagonals of Pascal's triangle at 45 degrees slope. - Javier Torres (adaycalledzero(AT)hotmail.com), Aug 06 2009
Let M be an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0, ...) in every column shifted down twice:
1;
1, 0;
1, 1, 0;
0, 1, 0, 0;
0, 1, 1, 0, 0;
0, 0, 1, 0, 0, 0;
0, 0, 1, 1, 0, 0, 0;
...
Then this sequence A002487 (without initial 0) is the first column of lim_{n->oo} M^n. (Cf. A026741.) - Gary W. Adamson, Dec 11 2009 [Edited by M. F. Hasler, Feb 12 2017]
Member of the infinite family of sequences of the form a(n) = a(2*n); a(2*n+1) = r*a(n) + a(n+1), r = 1 for A002487 = row 1 in the array of A178239. - Gary W. Adamson, May 23 2010
Equals row 1 in an infinite array shown in A178568, sequences of the form
a(2*n) = r*a(n), a(2*n+1) = a(n) + a(n+1); r = 1. - Gary W. Adamson, May 29 2010
Row sums of A125184, the Stern polynomials. Equivalently, B(n,1), the n-th Stern polynomial evaluated at x = 1. - T. D. Noe, Feb 28 2011
The Kn1y and Kn2y triangle sums, see A180662 for their definitions, of A047999 lead to the sequence given above, e.g., Kn11(n) = A002487(n+1) - A000004(n), Kn12(n) = A002487(n+3) - A000012(n), Kn13(n) = A002487(n+5) - A000034(n+1) and Kn14(n) = A002487(n+7) - A157810(n+1). For the general case of the knight triangle sums see the Stern-Sierpiński triangle A191372. This triangle not only leads to Stern's diatomic series but also to snippets of this sequence and, quite surprisingly, their reverse. - Johannes W. Meijer, Jun 05 2011
Maximum of terms between a(2^k) = 1 and a(2^(k+1)) = 1 is the Fibonacci number F(k+2). - Leonid Bedratyuk, Jul 04 2012
Probably the number of different entries per antidiagonal of A223541. That would mean there are exactly a(n+1) numbers that can be expressed as a nim-product 2^x*2^y with x + y = n. - Tilman Piesk, Mar 27 2013
Let f(m,n) be the frequency of the integer n in the interval [a(2^(m-1)), a(2^m-1)]. Let phi(n) be Euler's totient function (A000010). Conjecture: for all integers m,n n<=m f(m,n) = phi(n). - Yosu Yurramendi, Sep 08 2014
Back in May 1995, it was proved that A000360 is the modulo 3 mapping, (+1,-1,+0)/2, of this sequence A002487 (without initial 0). - M. Jeremie Lafitte (Levitas), Apr 24 2017
Define a sequence chf(n) of Christoffel words over an alphabet {-,+}: chf(1) = '-'; chf(2*n+0) = negate(chf(n)); chf(2*n+1) = negate(concatenate(chf(n),chf(n+1))). Then the length of the chf(n) word is fusc(n) = a(n); the number of '-'-signs in the chf(n) word is c-fusc(n) = A287729(n); the number of '+'-signs in the chf(n) word is s-fusc(n) = A287730(n). See examples below. - I. V. Serov, Jun 01 2017
The sequence can be extended so that a(n) = a(-n), a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1) for all n in Z. - Michael Somos, Jun 25 2019
Named after the German mathematician Moritz Abraham Stern (1807-1894), and sometimes also after the French clockmaker and amateur mathematician Achille Brocot (1817-1878). - Amiram Eldar, Jun 06 2021
It appears that a(n) is equal to the multiplicative inverse of A007305(n+1) mod A007306(n+1). For example, a(12) is 2, the multiplicative inverse of A007305(13) mod A007306(13), where A007305(13) is 4 and A007306(13) is 7. - Gary W. Adamson, Dec 18 2023

Examples

			Stern's diatomic array begins:
  1,1,
  1,2,1,
  1,3,2,3,1,
  1,4,3,5,2,5,3,4,1,
  1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,
  1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4,9,...
  ...
a(91) = 19, because 91_10 = 1011011_2; b_6=b_4=b_3=b_1=b_0=1, b_5=b_2=0;  L=5; m_1=0, m_2=1, m_3=3, m_4=4, m_5=6; c_1=2, c_2=3, c_3=2, c_4=3; f(1)=1, f(2)=2, f(3)=5, f(4)=8, f(5)=19. - _Yosu Yurramendi_, Jul 13 2016
From _I. V. Serov_, Jun 01 2017: (Start)
a(n) is the length of the Christoffel word chf(n):
n  chf(n) A070939(n)   a(n)
1   '-'       1          1
2   '+'       2          1
3   '+-'      2          2
4   '-'       3          1
5   '--+'     3          3
6   '-+'      3          2
... (End)
G.f. = x + x^2 + 2*x^3 + x^4 + 3*x^5 + 2*x^6 + 3*x^7 + x^8 + ... - _Michael Somos_, Jun 25 2019

References

  • M. Aigner and G. M. Ziegler, Proofs from THE BOOK, 3rd ed., Berlin, Heidelberg, New York: Springer-Verlag, 2004, p. 97.
  • Elwyn R. Berlekamp, John H. Conway and Richard K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 114.
  • Krishna Dasaratha, Laure Flapan, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse and Matthew Stroegeny, A family of multi-dimensional continued fraction Stern sequences, Abtracts Amer. Math. Soc., Vol. 33 (#1, 2012), #1077-05-2543.
  • Edsger W. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232 (sequence is called fusc).
  • F. G. M. Eisenstein, Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhaengen und durch gewisse lineare Funktional-Gleichungen definirt werden, Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin (1850), pp. 36-42, Feb 18, 1850. Werke, II, pp. 705-711.
  • Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.3; pp. 148-149.
  • Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117.
  • Thomas Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 98.
  • 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. A000120, A000123, A000360, A001045, A002083, A011655, A020950, A026741, A037227, A046815, A070871, A070872, A071883, A073459, A084091, A101624, A126606, A168081, A174980, A174981, A178239, A178568, A212288, A213369, A260443, A277020, A277325, A287729, A287730, A293160.
Record values are in A212289.
If the 1's are replaced by pairs of 1's we obtain A049456.
Inverse: A020946.
Cf. a(A001045(n)) = A000045(n). a(A062092(n)) = A000032(n+1).
Cf. A064881-A064886 (Stern-Brocot subtrees).
A column of A072170.
Cf. A049455 for the 0,1 version of Stern's diatomic array.
Cf. A000119, A262097 for analogous sequences in other bases and A277189, A277315, A277328 for related sequences with similar graphs.
Cf. A086592 and references therein to other sequences related to Kepler's tree of fractions.

Programs

  • Haskell
    a002487 n = a002487_list !! n
a002487_list = 0 : 1 : stern [1] where
   stern fuscs = fuscs' ++ stern fuscs' where
     fuscs' = interleave fuscs $ zipWith (+) fuscs $ (tail fuscs) ++ [1]
   interleave []     ys = ys
   interleave (x:xs) ys = x : interleave ys xs
-- Reinhard Zumkeller, Aug 23 2011
  • Julia
    using Nemo
function A002487List(len)
    a, A = QQ(0), [0,1]
    for n in 1:len
        a = next_calkin_wilf(a)
        push!(A, denominator(a))
    end
A end
A002487List(91) |> println # Peter Luschny, Mar 13 2018
  • Magma
    [&+[(Binomial(k, n-k-1) mod 2): k in [0..n]]: n in [0..100]]; // Vincenzo Librandi, Jun 18 2019
  • Maple
    A002487 := proc(n) option remember; if n <= 1 then n elif n mod 2 = 0 then procname(n/2); else procname((n-1)/2)+procname((n+1)/2); fi; end: seq(A002487(n),n=0..91);
A002487 := proc(m) 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(A002487(n),n=0..91); # Program adapted from E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232. - Igor Urbiha (urbiha(AT)math.hr), Oct 28 2002. Since A007306(n) = a(2*n+1), this program can be adapted for A007306 by replacing b := 0 by b := 1.
A002487 := proc(n::integer) local k; option remember; if n = 0 then 0 elif n=1 then 1 else add(K(k,n-1-k)*procname(n - k), k = 1 .. n) end if end proc:
K := proc(n::integer, k::integer) local KC; if 0 <= k and k <= n and n-k <= 2 then KC:=1; else KC:= 0; end if; end proc: seq(A002487(n),n=0..91); # Thomas Wieder, Jan 13 2008
# next Maple program:
a:= proc(n) option remember; `if`(n<2, n,
      (q-> a(q)+(n-2*q)*a(n-q))(iquo(n, 2)))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 11 2021
fusc := proc(n) local a, b, c; a := 1; b := 0;
    for c in convert(n, base, 2) do
        if c = 0 then a := a + b else b := a + b fi od;
    b end:
seq(fusc(n), n = 0..91); # Peter Luschny, Nov 09 2022
Stern := proc(n, u) local k, j, b;
    b := j -> nops({seq(Bits:-Xor(k, j-k), k = 0..j)}):
    ifelse(n=0, 1-u, seq(b(j), j = 2^(n-1)-1..2^n-1-u)) end:
seq(print([n], Stern(n, 1)), n = 0..5); # As shown in the comments.
seq(print([n], Stern(n, 0)), n = 0..5); # As shown in the examples. # Peter Luschny, Sep 29 2024
  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := If[ EvenQ[n], a[n/2], a[(n-1)/2] + a[(n+1)/2]]; Table[ a[n], {n, 0, 100}] (* end of program *)
Onemore[l_] := Transpose[{l, l + RotateLeft[l]}] // Flatten;
NestList[Onemore, {1}, 5] // Flatten  (*gives [a(1), ...]*) (* Takashi Tokita, Mar 09 2003 *)
ToBi[l_] := Table[2^(n - 1), {n, Length[l]}].Reverse[l]; Map[Length,
Split[Sort[Map[ToBi, Table[IntegerDigits[n - 1, 3], {n, 500}]]]]]  (*give [a(1), ...]*) (* Takashi Tokita, Mar 10 2003 *)
A002487[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b = a+b, a = a+b]; n = Floor[n/2]]; b]; Table[A002487[n], {n, 0, 100}] (* Jean-François Alcover, Sep 06 2013, translated from 2nd Maple program *)
a[0] = 0; a[1] = 1;
Flatten[Table[{a[2*n] = a[n], a[2*n + 1] = a[n] + a[n + 1]}, {n, 0, 50}]] (* Horst H. Manninger, Jun 09 2021 *)
nmax = 100; CoefficientList[Series[x*Product[(1 + x^(2^k) + x^(2^(k+1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2022 *)
  • PARI
    {a(n) = n=abs(n); if( n<2, n>0, a(n\2) + if( n%2, a(n\2 + 1)))};
  • PARI
    fusc(n)=local(a=1,b=0);while(n>0,if(bitand(n,1),b+=a,a+=b);n>>=1);b \\ Charles R Greathouse IV, Oct 05 2008
  • PARI
    A002487(n,a=1,b=0)=for(i=0,logint(n,2),if(bittest(n,i),b+=a,a+=b));b \\ M. F. Hasler, Feb 12 2017, updated Feb 14 2019
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def a(n): return n if n<2 else a(n//2) if n%2==0 else a((n - 1)//2) + a((n + 1)//2) # Indranil Ghosh, Jun 08 2017; corrected by Reza K Ghazi, Dec 27 2021
  • Python
    def a(n):
    a, b = 1, 0
    while n > 0:
        if n & 1:
            b += a
        else:
            a += b
        n >>= 1
    return b
# Reza K Ghazi, Dec 29 2021
  • Python
    def A002487(n): return sum(int(not (n-k-1) & ~k) for k in range(n)) # Chai Wah Wu, Jun 19 2022
  • Python
    # (fast way for big vectors)
from math import log, ceil
import numpy
how_many_terms = 2**20  # (Powers of 2 recommended but other integers are also possible.)
A002487, A002487[1]  = numpy.zeros(2**(ce:=ceil(log(how_many_terms,2))), dtype=object), 1
for exponent in range(1,ce):
    L, L2 = 2**exponent, 2**(exponent+1)
    A002487[L2 - 1] = exponent + 1
    A002487[L:L2][::2] = A002487[L >> 1: L]
    A002487[L + 1:L2 - 2][::2] = A002487[L:L2 - 3][::2]  +  A002487[L + 2:L2 - 1][::2]
print(list(A002487[0:100])) # Karl-Heinz Hofmann, Jul 22 2025
  • R
    N <- 50 # arbitrary
a <- 1
for (n in 1:N)
{
  a[2*n    ] = a[n]
  a[2*n + 1] = a[n] + a[n+1]
  a
}
a
# Yosu Yurramendi, Oct 04 2014
  • R
    # Given n, compute a(n) by taking into account the binary representation of n
a <- 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(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])
} # Yosu Yurramendi, Dec 13 2016
  • R
    # computes the sequence as a vector A, rather than function a() as above.
A <- c(1,1)
maxlevel <- 5 # by choice
for(m in 1:maxlevel) {
  A[2^(m+1)] <- 1
  for(k in 1:(2^m-1)) {
    r <- m - floor(log2(k)) - 1
    A[2^r*(2*k+1)] <- A[2^r*(2*k)] + A[2^r*(2*k+2)]
}}
A # Yosu Yurramendi, May 08 2018
  • Sage
    def A002487(n):
    M = [1, 0]
    for b in n.bits():
        M[b] = M[0] + M[1]
    return M[1]
print([A002487(n) for n in (0..91)])
# For a dual see A174980. Peter Luschny, Nov 28 2017
  • Scheme
    ;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
(definec (A002487 n) (cond ((<= n 1) n) ((even? n) (A002487 (/ n 2))) (else (+ (A002487 (/ (- n 1) 2)) (A002487 (/ (+ n 1) 2))))))
;; Antti Karttunen, Nov 05 2016

Formula

a(n+1) = (2*k+1)*a(n) - a(n-1) where k = floor(a(n-1)/a(n)). - David S. Newman, Mar 04 2001
Let e(n) = A007814(n) = exponent of highest power of 2 dividing n. Then a(n+1) = (2k+1)*a(n)-a(n-1), n > 0, where k = e(n). Moreover, floor(a(n-1)/a(n)) = e(n), in agreement with D. Newman's formula. - Dragutin Svrtan (dsvrtan(AT)math.hr) and Igor Urbiha (urbiha(AT)math.hr), Jan 10 2002
Calkin and Wilf showed 0.9588 <= limsup a(n)/n^(log(phi)/log(2)) <= 1.1709 where phi is the golden mean. Does this supremum limit = 1? - Benoit Cloitre, Jan 18 2004. Coons and Tyler show the limit is A246765 = 0.9588... - Kevin Ryde, Jan 09 2021
a(n) = Sum_{k=0..floor((n-1)/2)} (binomial(n-k-1, k) mod 2). - Paul Barry, Sep 13 2004
a(n) = Sum_{k=0..n-1} (binomial(k, n-k-1) mod 2). - Paul Barry, Mar 26 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 2*u*v*w - u^2*w. - Michael Somos, May 02 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^3*u6 - 3*u1^2*u2*u6 + 3*u2^3*u6 - u2^3*u3. - Michael Somos, May 02 2005
G.f.: x * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))) [Carlitz].
a(n) = a(n-2) + a(n-1) - 2*(a(n-2) mod a(n-1)). - Mike Stay, Nov 06 2006
A079978(n) = (1 + e^(i*Pi*A002487(n)))/2, i=sqrt(-1). - Paul Barry, Jan 14 2005
a(n) = Sum_{k=1..n} K(k, n-k)*a(n - k), where K(n,k) = 1 if 0 <= k AND k <= n AND n-k <= 2 and K(n,k) = 0 else. (When using such a K-coefficient, several different arguments to K or several different definitions of K may lead to the same integer sequence. For example, if we drop the condition k <= n in the above definition, then we arrive at A002083 = Narayana-Zidek-Capell numbers.) - Thomas Wieder, Jan 13 2008
a(k+1)*a(2^n - k) - a(k)*a(2^n - (k+1)) = 1; a(2^n - k) + a(k) = a(2^(n+1) + k). Both formulas hold for 0 <= k <= 2^n - 1. G.f.: G(z) = a(1) + a(2)*z + a(3)*z^2 + ... + a(k+1)*z^k + ... Define f(z) = (1 + z + z^2), then G(z) = lim f(z)*f(z^2)*f(z^4)* ... *f(z^(2^n))*... = (1 + z + z^2)*G(z^2). - Arie Werksma (werksma(AT)tiscali.nl), Apr 11 2008
a(k+1)*a(2^n - k) - a(k)*a(2^n - (k+1)) = 1 (0 <= k <= 2^n - 1). - Arie Werksma (werksma(AT)tiscali.nl), Apr 18 2008
a(2^n + k) = a(2^n - k) + a(k) (0 <= k <= 2^n). - Arie Werksma (werksma(AT)tiscali.nl), Apr 18 2008
Let g(z) = a(1) + a(2)*z + a(3)*z^2 + ... + a(k+1)*z^k + ..., f(z) = 1 + z + z^2. Then g(z) = lim_{n->infinity} f(z)*f(z^2)*f(z^4)*...*f(z^(2^n)), g(z) = f(z)*g(z^2). - Arie Werksma (werksma(AT)tiscali.nl), Apr 18 2008
For 0 <= k <= 2^n - 1, write k = b(0) + 2*b(1) + 4*b(2) + ... + 2^(n-1)*b(n-1) where b(0), b(1), etc. are 0 or 1. Define a 2 X 2 matrix X(m) with entries x(1,1) = x(2,2) = 1, x(1,2) = 1 - b(m), x(2,1) = b(m). Let P(n)= X(0)*X(1)* ... *X(n-1). The entries of the matrix P are members of the sequence: p(1,1) = a(k+1), p(1,2) = a(2^n - (k+1)), p(2,1) = a(k), p(2,2) = a(2^n - k). - Arie Werksma (werksma(AT)tiscali.nl), Apr 20 2008
Let f(x) = A030101(x); if 2^n + 1 <= x <= 2^(n + 1) and y = 2^(n + 1) - f(x - 1) then a(x) = a(y). - Arie Werksma (Werksma(AT)Tiscali.nl), Jul 11 2008
a(n) = A126606(n + 1) / 2. - Reikku Kulon, Oct 05 2008
Equals infinite convolution product of [1,1,1,0,0,0,0,0,0] aerated A000079 - 1 times, i.e., [1,1,1,0,0,0,0,0,0] * [1,0,1,0,1,0,0,0,0] * [1,0,0,0,1,0,0,0,1]. - Mats Granvik and Gary W. Adamson, Oct 02 2009; corrected by Mats Granvik, Oct 10 2009
a(2^(p+2)*n+2^(p+1)-1) - a(2^(p+1)*n+2^p-1) = A007306(n+1), p >= 0 and n >= 0. - Johannes W. Meijer, Feb 07 2013
a(2*n-1) = A007306(n), n > 0. - Yosu Yurramendi, Jun 23 2014
a(n*2^m) = a(n), m>0, n > 0. - Yosu Yurramendi, Jul 03 2014
a(k+1)*a(2^m+k) - a(k)*a(2^m+(k+1)) = 1 for m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Nov 07 2014
a(2^(m+1)+(k+1))*a(2^m+k) - a(2^(m+1)+k)*a(2^m+(k+1)) = 1 for m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Nov 07 2014
a(5*2^k) = 3. a(7*2^k) = 3. a(9*2^k) = 4. a(11*2^k) = 5. a(13*2^k) = 5. a(15*2^k) = 4. In general: a((2j-1)*2^k) = A007306(j), j > 0, k >= 0 (see Adamchuk's comment). - Yosu Yurramendi, Mar 05 2016
a(2^m+2^m'+k') = a(2^m'+k')*(m-m'+1) - a(k'), m >= 0, m' <= m-1, 0 <= k' < 2^m'. - Yosu Yurramendi, Jul 13 2016
From Yosu Yurramendi, Jul 13 2016: (Start)
Let n be a natural number and [b_m b_(m-1) ... b_1 b_0] its binary expansion with b_m=1.
Let L = Sum_{i=0..m} b_i be the number of binary digits equal to 1 (L >= 1).
Let {m_j: j=1..L} be the set of subindices such that b_m_j = 1, j=1..L, and 0 <= m_1 <= m_2 <= ... <= m_L = m.
If L = 1 then c_1 = 1, otherwise let {c_j: j=1..(L-1)} be the set of coefficients such that c_(j) = m_(j+1) - m_j + 1, 1 <= j <= L-1.
Let f be a function defined on {1..L+1} such that f(1) = 0, f(2) = 1, f(j) = c_(j-2)*f(j-1) - f(j-2), 3 <= j <= L+1.
Then a(n) = f(L+1) (see example). (End)
a(n) = A001222(A260443(n)) = A000120(A277020(n)). Also a(n) = A000120(A101624(n-1)) for n >= 1. - Antti Karttunen, Nov 05 2016
(a(n-1) + a(n+1))/a(n) = A037227(n) for n >= 1. - Peter Bala, Feb 07 2017
a(0) = 0; a(3n) = 2*A000360(3n-1); a(3n+1) = 2*A000360(3n) - 1; a(3n+2) = 2*A000360(3n+1) + 1. - M. Jeremie Lafitte (Levitas), Apr 24 2017
From I. V. Serov, Jun 14 2017: (Start)
a(n) = A287896(n-1) - 1*A288002(n-1) for n > 1;
a(n) = A007306(n-1) - 2*A288002(n-1) for n > 1. (End)
From Yosu Yurramendi, Feb 14 2018: (Start)
a(2^(m+2) + 2^(m+1) + k) - a(2^(m+1) + 2^m + k) = 2*a(k), m >= 0, 0 <= k < 2^m.
a(2^(m+2) + 2^(m+1) + k) - a(2^(m+1) + k) = a(2^m + k), m >= 0, 0 <= k < 2^m.
a(2^m + k) = a(k)*(m - floor(log_2(k)) - 1) + a(2^(floor(log_2(k))+1) + k), m >= 0, 0 < k < 2^m, a(2^m) = 1, a(0) = 0. (End)
From Yosu Yurramendi, May 08 2018: (Start)
a(2^m) = 1, m >= 0.
a(2^r*(2*k+1)) = a(2^r*(2*k)) + a(2^r*(2*k+2)), r < - m - floor(log_2(k)) - 1, m > 0, 1 <= k < 2^m. (End)
Trow(n) = [card({k XOR (j-k): k=0..j}) for j = 2^(n-1)-1..2^n-2] when regarded as an irregular table (n >= 1). - Peter Luschny, Sep 29 2024
a(n) = A000120(A168081(n)). - Karl-Heinz Hofmann, Jun 16 2025

Extensions

Additional references and comments from Len Smiley, Joshua Zucker, Rick L. Shepherd and Herbert S. Wilf
Typo in definition corrected by Reinhard Zumkeller, Aug 23 2011
Incorrect formula deleted and text edited by Johannes W. Meijer, Feb 07 2013

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

A277198 a(n) = gcd(A260443(n), A260443(n+1)).

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 15, 1, 1, 15, 15, 5, 5, 15, 105, 1, 1, 105, 75, 5, 5, 375, 525, 7, 7, 525, 525, 35, 35, 105, 1155, 1, 1, 1155, 525, 245, 35, 2625, 18375, 7, 7, 91875, 13125, 35, 245, 18375, 40425, 11, 11, 40425, 25725, 245, 245, 128625, 202125, 77, 77, 40425, 40425, 385, 385, 1155, 15015, 1, 1, 15015, 5775, 2695, 2695, 1414875, 1414875, 77, 77
Offset: 0

Views

Author

Antti Karttunen, Oct 10 2016

Keywords

Links

Crossrefs

Cf. A260443, A277197.
Cf. A277327, A277328 (number of prime factors).

Programs

  • PARI
    A=[];
A003961(n)=my(f=factor(n)); f[, 1] = apply(p->nextprime(p+1), f[, 1]); factorback(f)
A260443(n)=if(n<3, return(n+1)); if(#AA260443(n\2)*A260443(n\2+1), A003961(A260443(n/2)))
a(n)=gcd(A260443(n), A260443(n+1)) \\ Charles R Greathouse IV, Oct 13 2016
  • Scheme
    (define (A277198 n) (gcd (A260443 (+ 1 n)) (A260443 n)))
;; A more practical version, needing only an implementation of A000040:
(define (A277198 n) (product_primes_to_kth_powers (gcd_of_exp_lists (A260443as_coeff_list n) (A260443as_coeff_list (+ 1 n)))))
(define (product_primes_to_kth_powers nums) (let loop ((p 1) (nums nums) (i 1)) (cond ((null? nums) p) (else (loop (* p (expt (A000040 i) (car nums))) (cdr nums) (+ 1 i))))))
(definec (A260443as_coeff_list n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_coeff_list (/ n 2)))) (else (add_two_lists (A260443as_coeff_list (/ (- n 1) 2)) (A260443as_coeff_list (/ (+ n 1) 2))))))
(define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))
(define (gcd_of_exp_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (gcd_of_exp_lists nums2 nums1)) (else (map min nums1 (append nums2 (make-list (- len1 len2) 0)))))))

Formula

a(n) = gcd(A260443(n), A260443(n+1)).

A277327 Number of distinct primes dividing gcd(A260443(n), A260443(n+1)): a(n) = A001221(A277198(n)).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 0, 2, 2, 1, 1, 2, 3, 0, 0, 3, 2, 1, 1, 2, 3, 1, 1, 3, 3, 2, 2, 3, 4, 0, 0, 4, 3, 2, 2, 3, 3, 1, 1, 3, 3, 2, 2, 3, 4, 1, 1, 4, 3, 2, 2, 3, 4, 2, 2, 4, 4, 3, 3, 4, 5, 0, 0, 5, 4, 3, 3, 4, 4, 2, 2, 4, 3, 2, 2, 3, 4, 1, 1, 4, 3, 2, 2, 3, 4, 2, 2, 4, 4, 3, 3, 4, 5, 1, 1, 5, 4, 3, 3, 4, 4, 2, 2, 4, 4, 3, 3, 4, 5, 2, 2, 5, 4, 3, 3, 4, 5, 3, 3
Offset: 0

Views

Author

Antti Karttunen, Oct 13 2016

Keywords

Comments

a(n) = number of column positions where both row n and n+1 of A125184 have nonzero number present (when scanned from left), in other words, the number of k such that the term t^k has a nonzero coefficient in both Stern polynomials, B(n,t) and B(n+1,t).

Links

Crossrefs

Cf. A001221, A125184, A260443, A277198, A277328.

Programs

  • Scheme
    (define (A277327 n) (A001221 (A277198 n)))
;; A standalone implementation:
(define (A277327 n) (length (filter positive? (gcd_of_exp_lists (A260443as_coeff_list n) (A260443as_coeff_list (+ 1 n))))))
(definec (A260443as_coeff_list n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_coeff_list (/ n 2)))) (else (add_two_lists (A260443as_coeff_list (/ (- n 1) 2)) (A260443as_coeff_list (/ (+ n 1) 2))))))
(define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))
(define (gcd_of_exp_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (gcd_of_exp_lists nums2 nums1)) (else (map min nums1 (append nums2 (make-list (- len1 len2) 0)))))))

Formula

a(n) = A001221(A277198(n)).
a(n) <= A277328(n).

A284009 Number of primes (counted with multiplicity) dividing lcm(A260443(n), A260443(n+1)): a(n) = A001222(A284008(n)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 3, 4, 5, 5, 6, 6, 6, 6, 4, 5, 6, 6, 8, 9, 10, 9, 8, 8, 8, 8, 9, 9, 8, 8, 5, 6, 7, 7, 10, 10, 13, 13, 11, 12, 13, 12, 15, 16, 14, 13, 10, 10, 10, 10, 13, 14, 15, 14, 12, 12, 11, 11, 12, 12, 10, 10, 6, 7, 8, 8, 12, 12, 15, 15, 14, 15, 17, 17, 20, 21, 20, 19, 14, 15, 16, 15, 21, 22, 24, 24, 20, 21, 20, 18, 21, 21, 17, 17, 12, 12, 12, 12, 17, 18
Offset: 0

Views

Author

Antti Karttunen, Mar 22 2017

Keywords

Links

Crossrefs

Cf. A001222, A007306, A277328, A284008.

Programs

Formula

a(n) = A001222(A284008(n)).
Other identities. For all n >= 0:
a(n) + A277328(n) = A007306(1+n).

A278543 a(n) = A046523(A277198(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 6, 1, 1, 6, 6, 2, 2, 6, 30, 1, 1, 30, 12, 2, 2, 24, 60, 2, 2, 60, 60, 6, 6, 30, 210, 1, 1, 210, 60, 12, 6, 120, 360, 2, 2, 720, 240, 6, 12, 360, 1260, 2, 2, 1260, 360, 12, 12, 1080, 2520, 6, 6, 1260, 1260, 30, 30, 210, 2310, 1, 1, 2310, 420, 60, 60, 7560, 7560, 6, 6, 15120, 6480, 24, 24, 2160, 37800, 2, 2, 75600, 6480, 48, 48, 12960, 90720
Offset: 0

Views

Author

Antti Karttunen, Nov 30 2016

Keywords

Links

Crossrefs

Cf. A046523, A260443, A277198, A277327, A277328, A278243.

Programs

Formula

a(n) = A046523(A277198(n)).
Showing 1-6 of 6 results.

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