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A345699 Multiplicative with a(p) = a(p-1) and a(p^e) = a(p) + a(e) if e>1.

%I A345699 #28 Dec 12 2022 01:33:32
%S A345699 1,1,1,2,2,1,1,2,2,2,2,2,2,1,2,3,3,2,2,4,1,2,2,2,3,2,2,2,2,2,2,3,2,3,
%T A345699 2,4,4,2,2,4,4,1,1,4,4,2,2,3,2,3,3,4,4,2,4,2,2,2,2,4,4,2,2,2,4,2,2,6,
%U A345699 2,2,2,4,4,4,3,4,2,2,2,6,3,4,4,2,6,1,2
%N A345699 Multiplicative with a(p) = a(p-1) and a(p^e) = a(p) + a(e) if e>1.
%H A345699 Amiram Eldar, <a href="/A345699/b345699.txt">Table of n, a(n) for n = 1..10000</a>
%e A345699 a(11)=a(10)=a(5)*a(2); a(2)=1; a(5)=a(4)=a(2)+a(2)=2; so a(11)=2.
%p A345699 a:= proc(n) option remember; mul(`if`(i[2]=1,
%p A345699       a(i[1]-1), a(i[1])+a(i[2])), i=ifactors(n)[2])
%p A345699     end:
%p A345699 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 28 2021
%t A345699 a[1]=1; a[p_,1]:= a[p-1]; a[p_, s_] := a[p, s] =  a[p] + a[s];
%t A345699 a[n_]:=a[n]=Module[{aux=FactorInteger[n]},Product[a[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]];
%o A345699 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = a(f[k,1]-1); if (f[k,2] > 1, f[k,1] += a(f[k,2])); f[k,2] = 1); factorback(f); \\ _Michel Marcus_, Jun 26 2021
%Y A345699 Cf. A343068, A345763.
%K A345699 nonn,mult
%O A345699 1,4
%A A345699 _José María Grau Ribas_, Jun 24 2021