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.

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, 60, 41, 72, 31, 140, 33, 64, 65, 76, 63, 112, 39, 84, 75, 112, 43, 180, 45, 104, 98, 100, 49, 160, 58, 128, 95, 120, 55, 164

Offset: 1

Author

Yasutoshi Kohmoto, May 25 2005

Keywords

• nonn
• mult
• changed

Examples

			a(6) = (2+2)*(2+3) = 20.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 1..10000
• Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025.
• József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma+ function.

Crossrefs

Cf. A000203, A052396, A072691, A107759.

Programs

Maple

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:
seq(A107758(n),n=1..60) ; # R. J. Mathar, Jan 07 2011

Mathematica

Table[DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &], {n, 54}] (* Michael De Vlieger, Jun 27 2018 *)
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 *)

PARI

a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, sigma(d))); \\ Daniel Suteu, Jun 27 2018

Formula

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

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