A340324 -id:A340324 - cp's OEIS frontend

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

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, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54

Offset: 1

José María Grau Ribas, Jan 04 2021

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

			a(2^s) = 3 for all s>0.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021

fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]

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

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021

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

A340325 Numbers k such that starting with k and repeatedly applying the map x -> A340323(x) reaches the fixed point 12.

Original entry on oeis.org

5, 6, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84

Offset: 1

José María Grau Ribas, Jan 07 2021

nonn

From Sebastian Karlsson, Jan 15 2021: (Start)
The sequence contains no powers of two. If a number isn't a power of two, then it is in this sequence if and only if either of the following conditions hold:
- It is a multiple of a prime that is not a Mersenne prime.
- It is divisible by the square of a Mersenne prime greater than 3. (End)

Cf. A340323, A340324.

Cf. A000668.

fa[n_]:=fa[n]=FactorInteger[n];phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}];
S[n_] := NestWhile [phi, n, ! ( # == 12 || # == 3 || # == 4) &];
Select[1 + Range[100], S[#] == 12 &]

f(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)))); \\ A340323
isok(m) = if (m==1, return(0)); while(! ((m==3) || (m==4) || (m==12)), m = f(m)); (m==12); \\ Michel Marcus, Jan 21 2021

cp's OEIS Frontend

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

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