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A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.

%I A353789 #23 Dec 31 2022 03:52:34
%S A353789 1,4,12,24,30,48,70,144,180,120,132,288,208,280,360,864,306,720,418,
%T A353789 720,840,528,644,1728,1050,832,2700,1680,870,1440,1116,5184,1584,1224,
%U A353789 2100,4320,1480,1672,2496,4320,1722,3360,1978,3168,5400,2576,2444,10368,5390,4200,3672,4992,3074,10800,3960,10080,5016,3480
%N A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.
%C A353789 Question: Does a(n) divide A353790(n) only when n=1? Compare to A353764.
%H A353789 Antti Karttunen, <a href="/A353789/b353789.txt">Table of n, a(n) for n = 1..10000</a>
%H A353789 Antti Karttunen, <a href="/A353789/a353789.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A353789 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A353789 Multiplicative with a(p^e) = (q - 1) * q^(e-1) * p^e, where q is the least prime greater than p.
%F A353789 a(n) = A353749(A003961(n)) = n * A003972(n).
%F A353789 Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2-p+1)/(p^3 - p*q)) = 0.836506229..., where q(p) = nextprime(p) = A151800(p). - _Amiram Eldar_, Dec 31 2022
%t A353789 f[p_, e_] := ((q = NextPrime[p]) - 1) * q^(e - 1) * p^e; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 10 2022 *)
%o A353789 (PARI)
%o A353789 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A353789 A353789(n) = (n * eulerphi(A003961(n)));
%o A353789 (Python)
%o A353789 from math import prod
%o A353789 from sympy import nextprime, factorint
%o A353789 def A353789(n): return prod((q:= nextprime(p))**(e-1)*p**e*(q-1) for p, e in factorint(n).items()) # _Chai Wah Wu_, May 10 2022
%Y A353789 Cf. A003961, A003972, A151800, A353749, A353764, A353790.
%Y A353789 Cf. also A353793.
%K A353789 nonn,mult
%O A353789 1,2
%A A353789 _Antti Karttunen_, May 10 2022