cp's OEIS Frontend

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

%I A340323 #35 Nov 12 2022 05:25:38
%S A340323 1,3,4,3,6,12,8,3,8,18,12,12,14,24,24,3,18,24,20,18,32,36,24,12,24,42,
%T A340323 16,24,30,72,32,3,48,54,48,24,38,60,56,18,42,96,44,36,48,72,48,12,48,
%U A340323 72,72,42,54,48,72,24,80,90,60,72,62,96,64,3,84,144,68,54
%N A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).
%C A340323 Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.
%H A340323 Antti Karttunen, <a href="/A340323/b340323.txt">Table of n, a(n) for n = 1..20000</a>
%F A340323 a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - _Antti Karttunen_, Jan 06 2021
%F A340323 Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - _Amiram Eldar_, Nov 12 2022
%e A340323 a(2^s) = 3 for all s>0.
%p A340323 f:= proc(n) local  t;
%p A340323   mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
%p A340323 end proc:
%p A340323 map(f, [$1..100]); # _Robert Israel_, Jan 07 2021
%t A340323 fa[n_]:=fa[n]=FactorInteger[n];
%t A340323 phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
%t A340323 phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
%t A340323 Array[phi, 245]
%o A340323 (PARI) A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ _Antti Karttunen_, Jan 06 2021
%Y A340323 Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.
%K A340323 nonn,mult
%O A340323 1,2
%A A340323 _José María Grau Ribas_, Jan 04 2021