A167346 - OEIS

%I #17 Nov 05 2022 08:17:56

%S 1,4,10,16,28,40,54,64,100,112,130,160,180,216,280,256,304,400,378,

%T 448,540,520,550,640,784,720,1000,864,868,1120,990,1024,1300,1216,

%U 1512,1600,1404,1512,1800,1792,1720,2160,1890,2080,2800,2200,2254,2560,2916,3136

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

%H G. C. Greubel, <a href="/A167346/b167346.txt">Table of n, a(n) for n = 1..1000</a>

%F 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).

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

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 3)) = 1.611922780552146990915794949248803526278171368254928942581015265238806543... - _Vaclav Kotesovec_, Sep 20 2020

%F 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

%t 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 *)

%t 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 *)

%o (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

%Y Cf. A003958, A166590.

%K nonn,mult

%O 1,2

%A _Jaroslav Krizek_, Nov 01 2009