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

1,2

Links

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

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

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 *)

Prog

(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

Crossrefs

Cf. A003958, A166590.

Sequence in context: A343907 A191115 A073121 * A307274 A378708 A027430

Adjacent sequences: A167343 A167344 A167345 * A167347 A167348 A167349

Keyword

nonn,mult

Author

Jaroslav Krizek, Nov 01 2009

Status

approved