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A341528 a(n) = n * sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.

%I A341528 #21 Dec 08 2022 07:36:56
%S A341528 1,8,18,52,40,144,84,320,279,320,154,936,234,672,720,1936,340,2232,
%T A341528 456,2080,1512,1232,690,5760,1425,1872,4212,4368,928,5760,1178,11648,
%U A341528 2772,2720,3360,14508,1554,3648,4212,12800,1804,12096,2064,8008,11160,5520,2538,34848,6517,11400,6120,12168,3180,33696,6160,26880
%N A341528 a(n) = n * sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of the divisors of n.
%H A341528 Antti Karttunen, <a href="/A341528/b341528.txt">Table of n, a(n) for n = 1..8191</a>
%H A341528 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%H A341528 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A341528 Multiplicative with a(p^e) = (p^e) * (q^(e+1)-1)/(q-1) where q = nextPrime(p).
%F A341528 a(n) = n * A003973(n) = n * A000203(A003961(n)).
%F A341528 From _Antti Karttunen_, Mar 29 2021: (Start)
%F A341528 a(n) <= A341529(n).
%F A341528 a(n) = A341529(n) - A341512(n).
%F A341528 a(n) = A342662(A003961(n)).
%F A341528 (End)
%F A341528 Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 2.26342530..., where nextprime is A151800. - _Amiram Eldar_, Dec 08 2022
%t A341528 Array[#1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 56] (* _Michael De Vlieger_, Feb 22 2021 *)
%o A341528 (PARI)
%o A341528 A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A341528 A003973(n) = sigma(A003961(n));
%o A341528 A341528(n) = (n*A003973(n));
%Y A341528 Cf. A000203, A003961, A003973, A016754 (positions of the odd terms), A151800, A341512, A341526, A341527, A341529, A341530, A342661, A342662, A342673.
%K A341528 nonn,mult
%O A341528 1,2
%A A341528 _Antti Karttunen_, Feb 16 2021