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A107758 (+2)Sigma(n): If n = Product p_i^r_i then a(n) = Product (2 + Sum p_i^s_i, s_i=1 to r_i) = Product (1 + (p_i^(r_i+1)-1)/(p_i-1)), a(1) = 1.

%I A107758 #37 Aug 20 2025 10:48:09
%S A107758 1,4,5,8,7,20,9,16,14,28,13,40,15,36,35,32,19,56,21,56,45,52,25,80,32,
%T A107758 60,41,72,31,140,33,64,65,76,63,112,39,84,75,112,43,180,45,104,98,100,
%U A107758 49,160,58,128,95,120,55,164
%N A107758 (+2)Sigma(n): If n = Product p_i^r_i then a(n) = Product (2 + Sum p_i^s_i, s_i=1 to r_i) = Product (1 + (p_i^(r_i+1)-1)/(p_i-1)), a(1) = 1.
%H A107758 Daniel Suteu, <a href="/A107758/b107758.txt">Table of n, a(n) for n = 1..10000</a>
%H A107758 Sagar Mandal, <a href="https://arxiv.org/abs/2508.11660">Divisibility and Sequence Properties of sigma+ and phi+</a>, arXiv:2508.11660 [math.GM], 2025.
%H A107758 József Sándor and Krassimir Atanassov, <a href="https://doi.org/10.7546/nntdm.2024.30.4.851-856">Some new arithmetic functions</a>, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma+ function.
%F A107758 a(n) = Sum_{d|n, gcd(n/d, d) = 1} sigma(d), where sigma(d) is the sum of the divisors of d. - _Daniel Suteu_, Jun 27 2018
%F A107758 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.0741158... . - _Amiram Eldar_, Nov 01 2022
%e A107758 a(6) = (2+2)*(2+3) = 20.
%p A107758 A107758 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 1+(op(1,p)^(op(2,p)+1)-1)/(op(1,p)-1), p=pf) ; end if; end proc:
%p A107758 seq(A107758(n),n=1..60) ; # _R. J. Mathar_, Jan 07 2011
%t A107758 Table[DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &], {n, 54}] (* _Michael De Vlieger_, Jun 27 2018 *)
%t A107758 f[p_, e_] := 1 + (p^(e + 1) - 1)/(p - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Aug 26 2022 *)
%o A107758 (PARI) a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, sigma(d))); \\ _Daniel Suteu_, Jun 27 2018
%Y A107758 Cf. A000203, A052396, A072691, A107759.
%K A107758 nonn,mult,changed
%O A107758 1,2
%A A107758 _Yasutoshi Kohmoto_, May 25 2005