OFFSET
1,1
COMMENTS
Equivalently, with f = A008477, terms m of this sequence are precisely the nonsquarefree numbers for which the iterated sequence {m, f(m), f(f(m)), f(f(f(m))), ... } is not periodic.
There exist only these 4 possibilities:
-> for every squarefree number m in A005117, f(m) = 1, and iterated sequence is for example: (3, 1, 1, 1, 1, ...).
-> For m nonsquarefree fixed point of f in A008478, f(m) = m, iterated sequence has period = 1, as for example: (4, 4, 4, 4, 4, ...).
-> For m nonsquarefree in A062307, f(m) = q and f(q) = m, iterated sequence has period = 2, as for example: (8, 9, 8, 9, 8, 9, ...).
-> For m in this sequence, f(m) = k and m, k belong to an infinite iterated sequence, as for example: (..., 196, 512, 81, 64, ...) (see example).
EXAMPLE
MATHEMATICA
fun[p_, e_] := e^p; f[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[200], !SquareFreeQ[#] && f[#] != # && f[f[#]] != # &] (* Amiram Eldar, Apr 01 2021 *)
PROG
(PARI) f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477
isok(m) = if (!issquarefree(m), my(mm=f(m)); (mm != m) && (f(mm) != m)); \\ Michel Marcus, Apr 02 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Apr 01 2021
STATUS
approved