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%I #23 Mar 23 2021 16:16:48
%S 0,1,2,11,2,36,4,85,46,58,2,324,4,120,120,575,2,693,4,566,248,172,6,
%T 2340,94,270,788,1176,2,1800,6,3661,348,358,336,5967,4,492,548,4210,2,
%U 3744,4,1820,2490,744,6,15372,380,2271,720,2826,6,11304,392,8760,992,946,2,15480,6,1232,5164,22631,636,5904,4,3866
%N a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
%H Antti Karttunen, <a href="/A341512/b341512.txt">Table of n, a(n) for n = 1..8191</a>
%H Antti Karttunen, <a href="/A341512/a341512.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</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) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
%F For all primes p, a(p) = (q*(p+1)) - (p*(q+1)) = (pq + q) - (pq + p) = q - p = A001223(A000720(p)), where q = nextprime(p) = A003961(p).
%F And in general, a(p^e) = (q^e * (p^(e+1)-1)/(p-1)) - ((p^e) * (q^(e+1)-1)/(q-1)), where q = A003961(p).
%F Thus, a(p^2) = (p + 1)*q^2 - p^2*q - p^2,
%F a(p^3) = (p^2 + p + 1)*q^3 - p^3*q^2 - p^3*q - p^3,
%F a(p^4) = (p^3 + p^2 + p + 1)*q^4 - p^4*q^3 - p^4*q^2 - p^4*q - p^4,
%F etc.
%t Array[#2 DivisorSigma[1, #1] - #1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 68] (* _Michael De Vlieger_, Feb 22 2021 *)
%o (PARI)
%o A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A341528(n) = (n*sigma(A003961(n)));
%o A341529(n) = (sigma(n)*A003961(n));
%o A341512(n) = (A341529(n)-A341528(n));
%Y Cf. A000203, A000720, A001223, A003961, A003973, A286385, A341528, A341529, A341530.
%Y Cf. Sequences A001359, A029710, A031924 give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.
%K nonn,look
%O 1,3
%A _Antti Karttunen_, Feb 22 2021
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