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

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

Author

Yasutoshi Kohmoto, Jun 11 2005, Feb 24 2007

Keywords

• nonn
• easy
• mult

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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000079, A069184, A091321, A000203, A034448, A006519, A306633.

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