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

1,2

Comments

No prime numbers appear in this sequence.

Links

Table of n, a(n) for n=1..54.

Example

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)).

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 *)

Crossrefs

Cf. A056606, A056077.

Sequence in context: A365041 A144565 A334587 * A115912 A083096 A307348

Adjacent sequences: A329944 A329945 A329946 * A329948 A329949 A329950

Keyword

nonn

Author

Peter Luschny, Dec 21 2019

Status

approved