A167338 Totally multiplicative sequence with a(p) = p*(p+1) = p^2+p for prime p.

1, 6, 12, 36, 30, 72, 56, 216, 144, 180, 132, 432, 182, 336, 360, 1296, 306, 864, 380, 1080, 672, 792, 552, 2592, 900, 1092, 1728, 2016, 870, 2160, 992, 7776, 1584, 1836, 1680, 5184, 1406, 2280, 2184, 6480, 1722, 4032, 1892, 4752, 4320, 3312, 2256, 15552

Offset: 1

Author

Jaroslav Krizek, Nov 01 2009

Keywords

• nonn
• mult

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003959, A065489, A133205.

Programs

Mathematica

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*n, {n, 1, 100}] (* G. C. Greubel, Jun 06 2016 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + 1/(1/X/p - p - 1))/(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023

Formula

Multiplicative with a(p^e) = (p*(p+1))^e.

If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)+1))^e(k).

a(n) = n * A003959(n).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 1)) = A065489 = 1.419562880505485919317235861789735359166071586305122542698983695564330971... - Vaclav Kotesovec, Sep 20 2020

Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 2/p^2 - 1/p^3) = 0.8913709085... . - Amiram Eldar, Dec 15 2022, c = A065488/3. - Vaclav Kotesovec, Apr 05 2023

Dirichlet g.f.: zeta(s-2) * Product_{p prime} (1 + 1/(p^(s-1) - p - 1)). - Vaclav Kotesovec, Apr 05 2023