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

%I A329947 #13 Feb 28 2024 09:08:21
%S A329947 1,12,30,36,40,60,70,72,96,108,112,126,150,175,180,192,198,210,224,
%T A329947 240,270,280,306,312,324,330,336,350,351,352,378,384,396,399,400,408,
%U A329947 418,420,432,440,441,442,448,455,456,462,475,490,494,520,539,540,546,560
%N A329947 Integers k such that the radical of the cumulative product of k^k/k! equals its predecessor.
%C A329947 No prime numbers appear in this sequence.
%e A329947 Consider the rows 11 and 12 of Pascal's triangle.
%e A329947 P11 = [1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1].
%e A329947 P12 = [1, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1].
%e A329947 lcm(P11) = 2310 and radical(2310) = 2310.
%e A329947 lcm(P12) = 27720 and radical(27720) = 2310.
%e A329947 Since radical(lcm(P11)) = radical(lcm(P12)) 12 is in this sequence.
%e A329947 Also: 1 is in this sequence because radical(lcm(P0)) = radical(lcm([1])) = radical(1) = 1 = radical(lcm([1, 1])) = radical(lcm(P1)).
%p A329947 h := n -> mul(k^k/factorial(k), k=0..n):
%p A329947 rad := n -> mul(k, k = numtheory[factorset](n)):
%p A329947 g := proc(n) option remember; rad(h(n)) end:
%p A329947 isA329947 := n -> g(n) = g(n-1): select(isA329947, [$1..560]);
%t A329947 h[n_] := Product[k^k/k!, {k, 1, n}];
%t A329947 rad[n_] := Times @@ FactorInteger[n][[All, 1]];
%t A329947 g[n_] := g[n] = rad[h[n]];
%t A329947 isA329947[n_] := g[n] == g[n-1];
%t A329947 Select[Range[560], isA329947] (* _Jean-François Alcover_, Feb 28 2024, after Maple code *)
%Y A329947 Cf. A056606, A056077.
%K A329947 nonn
%O A329947 1,2
%A A329947 _Peter Luschny_, Dec 21 2019