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

Author

Jaroslav Krizek, Nov 01 2009

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A003958, A166591.

Programs

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 *)

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

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