A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..8191

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for primes, gaps between

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

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. A000203, A000720, A001223, A003961, A003973, A286385, A341528, A341529, A341530.

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

Adjacent sequences: A341509 A341510 A341511 * A341513 A341514 A341515

Keyword

nonn,look

Author

Antti Karttunen, Feb 22 2021

Status

approved