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%I #13 Dec 21 2023 10:22:42
%S 1,1,3,1,4,3,6,1,7,4,8,3,12,6,12,1,14,7,18,4,18,8,20,3,13,12,15,6,24,
%T 12,30,1,24,14,24,7,32,18,36,4,38,18,42,8,28,20,44,3,31,13,42,12,48,
%U 15,32,6,54,24,54,12,60,30,42,1,48,24,62,14,60,24,68,7,72,32,39,18,48,36,74,4,31,38,80,18,56,42,72,8
%N a(n) = sigma(A064989(n)).
%C The odd bisection is A285703, the even bisection is the sequence itself.
%H Antti Karttunen, <a href="/A326041/b326041.txt">Table of n, a(n) for n = 1..20000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (q^(e+1)-1)/(q-1), where q = A151799(p).
%F a(n) = A000203(A064989(n)).
%F a(2n) = a(n).
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/3) * Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.2722825585..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - _Amiram Eldar_, Dec 21 2023
%o (PARI) A326041(n) = if(1==n,n, my(f = factor(n)); prod(i=1, #f~, if(2==f[i,1],1,((precprime(f[i,1]-1)^(1+f[i,2]))-1)/(precprime(f[i,1]-1)-1))));
%o (PARI)
%o A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
%o A326041(n) = sigma(A064989(n));
%Y Cf. A000203, A003973, A064989, A151799, A285703.
%K nonn,mult
%O 1,3
%A _Antti Karttunen_, Jun 03 2019