A341528 - OEIS

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

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OFFSET

1,2

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).

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

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 *)

PROG

(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));

CROSSREFS

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

Sequence in context: A335440 A066721 A079704 * A032795 A120543 A337836

Adjacent sequences: A341525 A341526 A341527 * A341529 A341530 A341531

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Feb 16 2021

STATUS

approved