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A361266 Multiplicative with a(p^e) = p^(e + 3), e > 0.

%I A361266 #18 Sep 01 2023 04:09:46
%S A361266 1,16,81,32,625,1296,2401,64,243,10000,14641,2592,28561,38416,50625,
%T A361266 128,83521,3888,130321,20000,194481,234256,279841,5184,3125,456976,
%U A361266 729,76832,707281,810000,923521,256,1185921,1336336,1500625,7776,1874161,2085136,2313441,40000
%N A361266 Multiplicative with a(p^e) = p^(e + 3), e > 0.
%C A361266 In general, if the function is multiplicative with a(p^e) = p^(e + m) where m > 0, then Dirichlet g.f.: Product_{primes p} (1 + p^(m+1)/(p^s - p)).
%C A361266 Equivalently, Dirichlet g.f.: zeta(s-m-1) * zeta(s-1) * Product_{primes p} (1 + p^(2 + m - 2*s) - p^(2 + 2*m - 2*s) - p^(1 - s)).
%C A361266 Sum_{k=1..n} a(k) ~ c(m) * zeta(m+1) * n^(m+2) / (m+2), where c(m) = Product_{primes p} (1 - 1/p^2 - 1/p^(m+1) + 1/p^(m+2)).
%C A361266 Limit_{m->oo} c(m) = 6/Pi^2 = A059956.
%H A361266 Amiram Eldar, <a href="/A361266/b361266.txt">Table of n, a(n) for n = 1..10000</a>
%F A361266 Dirichlet g.f.: Product_{primes p} (1 + p^4/(p^s - p)).
%F A361266 Dirichlet g.f.: zeta(s-4) * zeta(s-1) * Product_{primes p} (1 + p^(5 - 2*s) - p^(8 - 2*s) - p^(1 - s)).
%F A361266 Sum_{k=1..n} a(k) ~ c * Pi^4 * n^5 / 450, where c = Product_{primes p} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.5761527353856670595206110782641172754062471168028961885...
%F A361266 From _Amiram Eldar_, Sep 01 2023: (Start)
%F A361266 a(n) = n * A007947(n)^3 = A064549(n) * A007947(n)^2 = A361264(n) * A007947(n) = A064549(A064549(A064549(n))).
%F A361266 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^3*(p-1))) = 1.148846213921... . (End)
%p A361266 f:= proc(n) local t;
%p A361266   mul(t[1]^(t[2]+3), t = ifactors(n)[2])
%p A361266 end proc:
%p A361266 map(f, [$1..50]); # _Robert Israel_, Mar 07 2023
%t A361266 g[p_, e_] := p^(e+3); a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
%o A361266 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + p^4*X/(1 - p*X))[n], ", "))
%o A361266 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,2] +=3); factorback(f); \\ _Michel Marcus_, Mar 07 2023
%Y A361266 Cf. A003557, A007947, A064549, A361264.
%K A361266 nonn,easy,mult
%O A361266 1,2
%A A361266 _Vaclav Kotesovec_, Mar 06 2023