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.

A107749 OrdinaryUnitarySigma(n): If n = Product p_i^r_i then OUSigma(n) = Sigma(2^r_1)*UnitarySigma(n/2^r_1).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 10, 18, 12, 28, 14, 24, 24, 31, 18, 30, 20, 42, 32, 36, 24, 60, 26, 42, 28, 56, 30, 72, 32, 63, 48, 54, 48, 70, 38, 60, 56, 90, 42, 96, 44, 84, 60, 72, 48, 124, 50, 78, 72, 98, 54, 84, 72, 120, 80, 90, 60, 168, 62, 96, 80, 127, 84, 144, 68, 126, 96
Offset: 1

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Author

Yasutoshi Kohmoto, Jun 11 2005, Feb 24 2007

Keywords

Comments

The sum of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079). - Amiram Eldar, Aug 26 2023

Examples

			OUSigma(2^4*7^2) = Sigma(2^4)*UnitarySigma(7^2) = 31*50 = 1550.
		

Crossrefs

Programs

  • Maple
    A107749 := proc(n) local a,f,p,e; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; if p = 2 then a := a*(2^(e+1)-1) ; else a := a*(p^e+1) ; end if; end do; a ; end proc: # R. J. Mathar, Jun 02 2011
  • Mathematica
    f[2, e_] := 2^(e+1)-1; f[p_, e_] := p^e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)
  • PARI
    a(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],if(fm[k,1]==2,2^(fm[k,2]+1)-1,fm[k,1]^fm[k,2]+1))

Formula

a(n) = A000203(p2) * A034448(n/p2), where p2 = A006519(n). - R. J. Mathar, Jun 15 2008
Multiplicative with a(2^e) = 2^(e+1)-1, a(p^e) = p^e+1 for p>2, e>0.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/7) * zeta(2)/zeta(3) = (4/7) * A306633 = 0.781961... . - Amiram Eldar, Nov 01 2022
Dirichlet g.f.: (4^s/(4^s-2)) * zeta(s)*zeta(s-1)/zeta(2*s-1). - Amiram Eldar, Aug 26 2023

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007
More terms from R. J. Mathar, Jun 15 2008
Name corrected by Franklin T. Adams-Watters, Aug 24 2013