A109624 Totally multiplicative sequence with a(p) = (p-1)*(p+3) = p^2+2p-3 for prime p.

1, 5, 12, 25, 32, 60, 60, 125, 144, 160, 140, 300, 192, 300, 384, 625, 320, 720, 396, 800, 720, 700, 572, 1500, 1024, 960, 1728, 1500, 896, 1920, 1020, 3125, 1680, 1600, 1920, 3600, 1440, 1980, 2304, 4000, 1760, 3600, 1932, 3500, 4608, 2860, 2300, 7500, 3600

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = ((p-1)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)+3))^e(k).

a(n) = A003958(n) * A166591(n).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 2*p - 4)) = 1.471999388763656342016756485604184156984049961181587531678650682804811302... - Vaclav Kotesovec, Sep 20 2020

Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 3/p^2 + 1/p^3 + 3/p^4)) = 0.6324191395... . - Amiram Eldar, Nov 05 2022

Mathematica

f[p_, e_] := ((p - 1)*(p + 3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2022 *)

Prog

(PARI) a(n) = {f = factor(n); return (prod(k=1, #f~, ((f[k, 1]-1)*(f[k, 1]+3))^f[k, 2])); } \\ Michel Marcus, Jun 13 2013

Crossrefs

Cf. A003958, A166591.

Sequence in context: A079425 A272194 A169699 * A081501 A355947 A274629

Adjacent sequences: A109621 A109622 A109623 * A109625 A109626 A109627

Keyword

nonn,mult

Author

Jaroslav Krizek, Nov 01 2009

Status

approved