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A341512
a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
11
0, 1, 2, 11, 2, 36, 4, 85, 46, 58, 2, 324, 4, 120, 120, 575, 2, 693, 4, 566, 248, 172, 6, 2340, 94, 270, 788, 1176, 2, 1800, 6, 3661, 348, 358, 336, 5967, 4, 492, 548, 4210, 2, 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
OFFSET
1,3
FORMULA
a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
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).
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).
Thus, a(p^2) = (p + 1)*q^2 - p^2*q - p^2,
a(p^3) = (p^2 + p + 1)*q^3 - p^3*q^2 - p^3*q - p^3,
a(p^4) = (p^3 + p^2 + p + 1)*q^4 - p^4*q^3 - p^4*q^2 - p^4*q - p^4,
etc.
MATHEMATICA
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 *)
PROG
(PARI)
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));
CROSSREFS
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.
Sequence in context: A037299 A329941 A347121 * A276676 A077805 A344539
KEYWORD
nonn,look
AUTHOR
Antti Karttunen, Feb 22 2021
STATUS
approved