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.

1, 8, 18, 52, 40, 144, 84, 320, 279, 320, 154, 936, 234, 672, 720, 1936, 340, 2232, 456, 2080, 1512, 1232, 690, 5760, 1425, 1872, 4212, 4368, 928, 5760, 1178, 11648, 2772, 2720, 3360, 14508, 1554, 3648, 4212, 12800, 1804, 12096, 2064, 8008, 11160, 5520, 2538, 34848, 6517, 11400, 6120, 12168, 3180, 33696, 6160, 26880

Offset: 1

Author

Antti Karttunen, Feb 16 2021

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8191
• Index entries for sequences computed from indices in prime factorization.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A003961, A003973, A016754 (positions of the odd terms), A151800, A341512, A341526, A341527, A341529, A341530, A342661, A342662, A342673.

Programs

Mathematica

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 *)

PARI

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A003973(n) = sigma(A003961(n));
A341528(n) = (n*A003973(n));

Formula

Multiplicative with a(p^e) = (p^e) * (q^(e+1)-1)/(q-1) where q = nextPrime(p).

a(n) = n * A003973(n) = n * A000203(A003961(n)).

From Antti Karttunen, Mar 29 2021: (Start)

a(n) <= A341529(n).

a(n) = A341529(n) - A341512(n).

a(n) = A342662(A003961(n)).

(End)

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