A167346 Totally multiplicative sequence with a(p) = (p-1)*(p+2) = p^2+p-2 for prime p.

1, 4, 10, 16, 28, 40, 54, 64, 100, 112, 130, 160, 180, 216, 280, 256, 304, 400, 378, 448, 540, 520, 550, 640, 784, 720, 1000, 864, 868, 1120, 990, 1024, 1300, 1216, 1512, 1600, 1404, 1512, 1800, 1792, 1720, 2160, 1890, 2080, 2800, 2200, 2254, 2560, 2916, 3136

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. A003958, A166590.

Programs

Mathematica

a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)
f[p_, e_] := ((p - 1)*(p + 2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2022 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, ((f[i,1]-1)*(f[i,1]+2))^f[i,2]); } \\ Amiram Eldar, Nov 05 2022

Formula

Multiplicative with a(p^e) = ((p-1)*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)+2))^e(k).

a(n) = A003958(n) * A166590(n).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 3)) = 1.611922780552146990915794949248803526278171368254928942581015265238806543... - 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 + 2/p^4)) = 0.3809790887... . - Amiram Eldar, Nov 05 2022