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Totally multiplicative sequence with a(p) = (p+2)*(p+3) = p^2+5p+6 for prime p.
1

%I #12 Sep 20 2020 04:43:17

%S 1,20,30,400,56,600,90,8000,900,1120,182,12000,240,1800,1680,160000,

%T 380,18000,462,22400,2700,3640,650,240000,3136,4800,27000,36000,992,

%U 33600,1122,3200000,5460,7600,5040,360000,1560,9240,7200,448000,1892,54000,2070

%N Totally multiplicative sequence with a(p) = (p+2)*(p+3) = p^2+5p+6 for prime p.

%H G. C. Greubel, <a href="/A167360/b167360.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = ((p+2)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+2)*(p(k)+3))^e(k). a(n) = A166590(n) * A166591(n).

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 5*p + 5)) = 1.1480407951783735490090642594369977652983537687209929674246821640934042061... - _Vaclav Kotesovec_, Sep 20 2020

%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* _G. C. Greubel_, Jun 11 2016 *)

%Y Cf. A166590, A166591.

%K nonn,mult

%O 1,2

%A _Jaroslav Krizek_, Nov 01 2009