A167351 Totally multiplicative sequence with a(p) = (p+1)*(p+2) = p^2+3p+2 for prime p.

1, 12, 20, 144, 42, 240, 72, 1728, 400, 504, 156, 2880, 210, 864, 840, 20736, 342, 4800, 420, 6048, 1440, 1872, 600, 34560, 1764, 2520, 8000, 10368, 930, 10080, 1056, 248832, 3120, 4104, 3024, 57600, 1482, 5040, 4200, 72576, 1806, 17280, 1980, 22464

Offset: 1

Author

Jaroslav Krizek, Nov 01 2009

Keywords

• nonn
• mult

Links

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

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 *)

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) = A003959(n) * A166590(n).

Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 3*p + 1)) = 1.224476389903759550811745481197762941643093896189832037452375111814242433... - Vaclav Kotesovec, Sep 20 2020