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.

Showing 1-5 of 5 results.

A092150 Partial sums of A092674.

Original entry on oeis.org

0, 3, 6, 7, 12, 12, 19, 23, 29, 31, 42, 47, 60, 64, 71, 79, 96, 102, 121, 130, 141, 149, 172, 180, 200, 210, 228, 241, 270, 280, 311, 327, 346, 360, 383, 395, 432, 448, 471, 487, 528, 542, 585, 606, 630, 650, 697, 713, 755, 775, 806, 831, 884, 902, 941, 965, 1000
Offset: 1

Views

Author

Jon Perry, Mar 31 2004

Keywords

A092151 A092673(n)*A092674(n).

Original entry on oeis.org

0, -6, -3, 1, -5, 0, -7, 0, 0, 4, -11, -5, -13, 8, 7, 0, -17, 0, -19, -9, 11, 16, -23, 0, 0, 20, 0, -13, -29, -20, -31, 0, 19, 28, 23, 0, -37, 32, 23, 0, -41, -28, -43, -21, 0, 40, -47, 0, 0, 0
Offset: 1

Views

Author

Jon Perry, Mar 31 2004

Keywords

A092154 First differences of A092674.

Original entry on oeis.org

3, 0, -2, 4, -5, 7, -3, 2, -4, 9, -6, 8, -9, 3, 1, 9, -11, 13, -10, 2, -3, 15, -15, 12, -10, 8, -5, 16, -19, 21, -15, 3, -5, 9, -11, 25, -21, 7, -7, 25, -27, 29, -22, 3, -4, 27, -31, 26, -22, 11, -6, 28, -35, 21, -15, 11, -9, 33, -44, 46, -33, 8, -4, 15, -25, 45, -34, 10, -17, 45, -47, 49, -39, 6, -3, 22
Offset: 1

Views

Author

Jon Perry, Mar 31 2004

Keywords

A092169 A000217(n)-A092674(n).

Original entry on oeis.org

1, 0, 3, 9, 10, 21, 21, 32, 39, 53, 55, 73, 78, 101, 113, 128, 136, 165, 171, 201, 220, 245, 253, 292, 305, 341, 360, 393, 406, 455, 465, 512, 542, 581, 607, 654, 666, 725, 757, 804, 820, 889, 903, 969, 1011, 1061, 1081, 1160, 1183, 1255, 1295, 1353, 1378, 1467
Offset: 1

Views

Author

Jon Perry, Apr 01 2004

Keywords

A092673 a(n) = moebius(n) - moebius(n/2) where moebius(n) is zero if n is not an integer.

Original entry on oeis.org

1, -2, -1, 1, -1, 2, -1, 0, 0, 2, -1, -1, -1, 2, 1, 0, -1, 0, -1, -1, 1, 2, -1, 0, 0, 2, 0, -1, -1, -2, -1, 0, 1, 2, 1, 0, -1, 2, 1, 0, -1, -2, -1, -1, 0, 2, -1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 1, 2, -1, 1, -1, 2, 0, 0, 1, -2, -1, -1, 1, -2, -1, 0, -1, 2, 0, -1, 1, -2, -1, 0, 0, 2, -1, 1, 1, 2, 1, 0, -1, 0, 1, -1, 1, 2, 1, 0, -1, 0, 0, 0, -1, -2, -1, 0, -1, 2
Offset: 1

Views

Author

Jon Perry, Mar 02 2004

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Setting x=1 gives us phi(n) (A000010). Setting x=0 gives A092674.
Apparently the Dirichlet inverse of A001511. - R. J. Mathar, Dec 22 2010
Given A = A115359 as an infinite lower triangular matrix and B = the Mobius sequence as a vector, A092673 = A*B. - Gary W. Adamson, Mar 14 2011
Empirical: Letting M(n) denote the n X n matrix whereby the (i,j)-entry of M(n) is Sum_{k=1..j} floor(i/k), we have that a(n) is the (n,1)-entry of the inverse of M(n). - John M. Campbell, Aug 30 2017
John Campbell's statement is proved at the Mathematics Stack Exchange link. - Sungjin Kim, Jul 17 2019

Examples

			The first few s[n] are:
x, -2*x + 3, -x + 3, x + 1, -x + 5, 2*x, -x + 7, 4, 6, 2*x + 2, -x + 11, -x + 5, -x + 13, 2*x + 4, x + 7, 8, -x + 17, 6, -x + 19, -x + 9, x + 11, 2*x + 8, -x + 23, 8, 20, 2*x + 10, 18, -x + 13, -x + 29, -2*x + 10, -x + 31, 16, x + 19.
x - 2*x^2 - x^3 + x^4 - x^5 + 2*x^6 - x^7 + 2*x^10 - x^11 +...

Links

Crossrefs

Cf. A008683 (moebius(n)), A092149 (partial sums), A092674, A115359.
Cf. A000010, A000122, A000700, A010054, A092673, A115359, A121373.

Programs

  • Maple
    A092673:= proc(n) if n::odd then numtheory:-mobius(n) else numtheory:-mobius(n) - numtheory:-mobius(n/2) fi end proc:
map(A092673, [$1..100]); # Robert Israel, Dec 31 2015
  • Mathematica
    f[n_] := MoebiusMu[n] - If[OddQ@n, 0, MoebiusMu[n/2]]; Array[f, 105] (* Robert G. Wilson v *)
  • PARI
    s=vector(2000); t(n)=binomial(n+1,2); s[1]=x; for(i=2,2000, s[i]=t(i)-sum(j=1,i-1, s[j]*floor(i/j))); for(i=1,2000,print1(","polcoeff(s[i],1)))
  • PARI
    {a(n) = if( n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2)))} /* Michael Somos, Mar 26 2007 */
  • PARI
    {a(n) = local(A, B, m); if( n<1, 0, A = x * O(x^n); B = 1 + x + A; for( k=1, n, B *= eta(x^k + A)^( m = polcoeff(B, k))); m)} /* Michael Somos, Mar 26 2007 */
  • PARI
    a(n)=my(o=valuation(n%8,2)); if(o==0, moebius(n), if(o==1, 2*moebius(n), if(o==2, moebius(n/4), 0))) \\ Charles R Greathouse IV, Feb 07 2013
  • Python
    from sympy import mobius
def A092673(n): return mobius(n)-(0 if n&1 else mobius(n>>1)) # Chai Wah Wu, Jul 13 2022

Formula

Let t(n) = binomial(n+1,2); s[1]=x; for i >= 2, s[i] = t(i)-Sum_{j=1..i-1} s[j]*floor(i/j); a(n) = coefficient of x in s[n]. - Jon Perry
a(n) is multiplicative with a(2)= -2, a(4)= 1, a(2^e)= 0 if e>2. a(p)= -1, a(p^e)= 0 if e>1, p>2. - Michael Somos, Mar 26 2007
a(8*n)= 0. a(2*n + 1) = moebius(2*n + 1). a(2*n) = moebius(2*n) - moebius(n). - Michael Somos, Mar 26 2007
|a(n)| <= 2.
1 / (1 + x) = Product_{k>0} f(-x^k)^a(k) where f() is a Ramanujan theta function. - Michael Somos, Mar 26 2007
Dirichlet g.f.: (1-2^(-s))/zeta(s). - Ralf Stephan, Mar 24 2015
G.f. A(x) satisfies x * (1 - x) = Sum_{k>=1} A(x^k). - Seiichi Manyama, Mar 31 2023
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 9/Pi^2 = 0.9118906... . - Amiram Eldar, Jan 19 2024
Showing 1-5 of 5 results.

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