A342973 Let f = A008477; nonsquarefree numbers m such that f(m) <> m or f(f(m)) <> m.

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192

Offset: 1

Author

Bernard Schott, Apr 01 2021

Keywords

• nonn

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.

The first sixteen terms are the same as A126706, then a(17) = 64 while A126706(17) = 68.

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).

Examples

			196 = 2^2*7^2 => A008477(196) = 2^2*2^7 = 2^9 = 512.
512 = 2^9 => A008477(512) = 9^2 = 81.
81 = 9^2 = 3^4 => A008477(81) = 4^3 = 64.
196, 512, 81, 64 are not terms of (A008478 U A062307), so they belong to this sequence.
		

Crossrefs

Equals A013929 \ {A008478 U A062307}.

Cf. A008477, A342551.

Programs

Mathematica

fun[p_, e_] := e^p; f[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[200], !SquareFreeQ[#] && f[#] != # && f[f[#]] != # &] (* Amiram Eldar, Apr 01 2021 *)

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