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%I #36 Aug 26 2023 03:51:56
%S 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,
%T 42,28,56,30,72,32,63,48,54,48,70,38,60,56,90,42,96,44,84,60,72,48,
%U 124,50,78,72,98,54,84,72,120,80,90,60,168,62,96,80,127,84,144,68,126,96
%N OrdinaryUnitarySigma(n): If n = Product p_i^r_i then OUSigma(n) = Sigma(2^r_1)*UnitarySigma(n/2^r_1).
%C The sum of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079). - _Amiram Eldar_, Aug 26 2023
%H Antti Karttunen, <a href="/A107749/b107749.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000203(p2) * A034448(n/p2), where p2 = A006519(n). - _R. J. Mathar_, Jun 15 2008
%F Multiplicative with a(2^e) = 2^(e+1)-1, a(p^e) = p^e+1 for p>2, e>0.
%F 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
%F Dirichlet g.f.: (4^s/(4^s-2)) * zeta(s)*zeta(s-1)/zeta(2*s-1). - _Amiram Eldar_, Aug 26 2023
%e OUSigma(2^4*7^2) = Sigma(2^4)*UnitarySigma(7^2) = 31*50 = 1550.
%p 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
%t 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 *)
%o (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))
%Y Cf. A000079, A069184, A091321, A000203, A034448, A006519, A306633.
%K nonn,easy,mult
%O 1,2
%A _Yasutoshi Kohmoto_, Jun 11 2005, Feb 24 2007
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 08 2007
%E More terms from _R. J. Mathar_, Jun 15 2008
%E Name corrected by _Franklin T. Adams-Watters_, Aug 24 2013
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