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%I #19 Dec 24 2023 01:09:11
%S 1,8,19,54,41,152,87,342,305,328,155,1026,237,696,779,2106,341,2440,
%T 459,2214,1653,1240,695,6498,1477,1896,4675,4698,929,6232,1183,12798,
%U 2945,2728,3567,16470,1557,3672,4503,14022,1805,13224,2067,8370,12505,5560,2543,40014,6809,11816,6479,12798,3185,37400,6355
%N Möbius transform of A341529, sigma(n) * A003961(n).
%C Multiplicative because A341529 is.
%H Antti Karttunen, <a href="/A347125/b347125.txt">Table of n, a(n) for n = 1..10000</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 a(n) = Sum_{d|n} A008683(n/d) * A341529(d).
%F a(n) = A346239(n) + A347124(n).
%F Multiplicative with a(p^e) = q^(e-1)*(p^e*(q*p-1)-q+1)/(p-1), where q = A151800(p). - _Sebastian Karlsson_, Sep 02 2021
%F Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = (1/zeta(3)) / Product_{p prime} ((p^2-q)*(p^3-q))/(p^4*(p-1)) = 7.6530842... , and q(p) = A151800(p). - _Amiram Eldar_, Dec 24 2023
%t f[p_, e_] := Module[{q = NextPrime[p]}, q^(e-1) * (p^e * (q*p-1)-q+1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 24 2023 *)
%o (PARI)
%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A341529(n) = (sigma(n)*A003961(n));
%o A347125(n) = sumdiv(n,d,moebius(n/d)*A341529(d))
%Y Cf. A002117, A003961, A003973, A008683, A151800, A341529, A341512, A346239, A347124.
%K nonn,mult
%O 1,2
%A _Antti Karttunen_, Aug 24 2021
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