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

%I #28 Apr 05 2023 08:46:55

%S 1,6,12,36,30,72,56,216,144,180,132,432,182,336,360,1296,306,864,380,

%T 1080,672,792,552,2592,900,1092,1728,2016,870,2160,992,7776,1584,1836,

%U 1680,5184,1406,2280,2184,6480,1722,4032,1892,4752,4320,3312,2256,15552

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

%H G. C. Greubel, <a href="/A167338/b167338.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = (p*(p+1))^e.

%F If n = Product p(k)^e(k) then a(n) = Product (p(k)*(p(k)+1))^e(k).

%F a(n) = n * A003959(n).

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 1)) = A065489 = 1.419562880505485919317235861789735359166071586305122542698983695564330971... - _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) = 0.8913709085... . - _Amiram Eldar_, Dec 15 2022, c = A065488/3. - _Vaclav Kotesovec_, Apr 05 2023

%F Dirichlet g.f.: zeta(s-2) * Product_{p prime} (1 + 1/(p^(s-1) - p - 1)). - _Vaclav Kotesovec_, Apr 05 2023

%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); Table[a[n]*n, {n, 1, 100}] (* _G. C. Greubel_, Jun 06 2016 *)

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 1/(1/X/p - p - 1))/(1 - p^2*X))[n], ", ")) \\ _Vaclav Kotesovec_, Apr 05 2023

%Y Cf. A003959, A065489, A133205.

%K nonn,mult

%O 1,2

%A _Jaroslav Krizek_, Nov 01 2009