A329947 Integers k such that the radical of the cumulative product of k^k/k! equals its predecessor.

1, 12, 30, 36, 40, 60, 70, 72, 96, 108, 112, 126, 150, 175, 180, 192, 198, 210, 224, 240, 270, 280, 306, 312, 324, 330, 336, 350, 351, 352, 378, 384, 396, 399, 400, 408, 418, 420, 432, 440, 441, 442, 448, 455, 456, 462, 475, 490, 494, 520, 539, 540, 546, 560

Offset: 1

Author

Peter Luschny, Dec 21 2019

Keywords

• nonn

Comments

No prime numbers appear in this sequence.

Examples

			Consider the rows 11 and 12 of Pascal's triangle.
P11 = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1].
P12 = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1].
lcm(P11) = 2310 and radical(2310) = 2310.
lcm(P12) = 27720 and radical(27720) = 2310.
Since radical(lcm(P11)) = radical(lcm(P12)) 12 is in this sequence.
Also: 1 is in this sequence because radical(lcm(P0)) = radical(lcm([1])) = radical(1) = 1 = radical(lcm([1, 1])) = radical(lcm(P1)).
		

Crossrefs

Cf. A056606, A056077.

Programs

Maple

h := n -> mul(k^k/factorial(k), k=0..n):
rad := n -> mul(k, k = numtheory[factorset](n)):
g := proc(n) option remember; rad(h(n)) end:
isA329947 := n -> g(n) = g(n-1): select(isA329947, [$1..560]);

Mathematica

h[n_] := Product[k^k/k!, {k, 1, n}];
rad[n_] := Times @@ FactorInteger[n][[All, 1]];
g[n_] := g[n] = rad[h[n]];
isA329947[n_] := g[n] == g[n-1];
Select[Range[560], isA329947] (* Jean-François Alcover, Feb 28 2024, after Maple code *)