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A343068 Multiplicative with a(p^e) = e*a(p-1).

%I A343068 #40 Dec 12 2022 01:41:31
%S A343068 1,1,1,2,2,1,1,3,2,2,2,2,2,1,2,4,4,2,2,4,1,2,2,3,4,2,3,2,2,2,2,5,2,4,
%T A343068 2,4,4,2,2,6,6,1,1,4,4,2,2,4,2,4,4,4,4,3,4,3,2,2,2,4,4,2,2,6,4,2,2,8,
%U A343068 2,2,2,6,6,4,4,4,2,2,2,8,4,6,6,2,8,1
%N A343068 Multiplicative with a(p^e) = e*a(p-1).
%H A343068 Amiram Eldar, <a href="/A343068/b343068.txt">Table of n, a(n) for n = 1..10000</a>
%F A343068 a(2^n) = n*a(2-1) = n.
%e A343068 a(2) = a(2-1) = 1; a(3) = a(3-1) = 1.
%p A343068 a:= proc(n) option remember; `if`(n=1, 1,
%p A343068       mul(i[2]*a(i[1]-1), i=ifactors(n)[2]))
%p A343068     end:
%p A343068 seq(a(n), n=1..100);  # _Alois P. Heinz_, Apr 04 2021
%t A343068 a[1] = 1; a[p_,s_]:= a[p, s]=s a[p-1];
%t A343068 a[n_]:=a[n]= Module[{aux = FactorInteger[n]},Product[a[aux[[i, 1]],aux[[i, 2]]], {i, Length[aux]}]]; Table[a[n],{n,100}]
%o A343068 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f[k,2]*a(f[k,1]-1); f[k,2] = 1); factorback(f); \\ _Michel Marcus_, Apr 23 2021
%Y A343068 Cf. A279513, A191614.
%K A343068 nonn,mult
%O A343068 1,4
%A A343068 _José María Grau Ribas_, Apr 04 2021