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Powerful(1) numbers (A001694) that are sums of distinct factorials.
3

%I #10 Feb 24 2024 11:04:40

%S 1,8,9,25,27,32,121,128,144,729,841,864,5041,5184,40328,41067,45369,

%T 45387,46208,46225,363609,403225,3674889,43954688,6230694987,

%U 1401602635449

%N Powerful(1) numbers (A001694) that are sums of distinct factorials.

%C Factorials 0! and 1! are not considered distinct.

%C a(27) > 10^18, if it exists. - _Amiram Eldar_, Feb 24 2024

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%e 6230694987 = 13!+10!+8!+7!+4!+2!+1! = 3^3*11^2*1381^2.

%t pwfQ[n_] := n==1 || Min[Last /@ FactorInteger@n] > 1; fac=Range[20]!;lst={}; Do[ n = Plus@@(fac*IntegerDigits[k, 2, 20]); If[pwfQ[n], AppendTo[lst, n]], {k, 2^20-1}]; lst

%t q[n_] := Module[{k = n, m = 2, r, ans = True}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 1, ans = False; Break[]]; m++]; ans]; With[{max = 2^20-1}, Select[Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]], q]] (* _Amiram Eldar_, Feb 24 2024 *)

%Y Intersection of A001694 and A059590.

%Y Cf. A025494, A115644, A115646, A089359.

%K nonn,more

%O 1,2

%A _Giovanni Resta_, Jan 27 2006