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%I #29 Nov 01 2022 07:14:02
%S 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 60,41,72,31,140,33,64,65,76,63,112,39,84,75,112,43,180,45,104,98,100,
%U 49,160,58,128,95,120,55,164
%N (+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 Daniel Suteu, <a href="/A107758/b107758.txt">Table of n, a(n) for n = 1..10000</a>
%F 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 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 a(6) = (2+2)*(2+3) = 20.
%p 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 seq(A107758(n),n=1..60) ; # _R. J. Mathar_, Jan 07 2011
%t Table[DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &], {n, 54}] (* _Michael De Vlieger_, Jun 27 2018 *)
%t 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 (PARI) a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, sigma(d))); \\ _Daniel Suteu_, Jun 27 2018
%Y Cf. A000203, A052396, A072691, A107759.
%K nonn,mult
%O 1,2
%A _Yasutoshi Kohmoto_, May 25 2005
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