A115645 Powerful(1) numbers (A001694) that are sums of distinct factorials.

1, 8, 9, 25, 27, 32, 121, 128, 144, 729, 841, 864, 5041, 5184, 40328, 41067, 45369, 45387, 46208, 46225, 363609, 403225, 3674889, 43954688, 6230694987, 1401602635449

Offset: 1

Author

Giovanni Resta, Jan 27 2006

Keywords

• nonn
• more

Comments

Factorials 0! and 1! are not considered distinct.

a(27) > 10^18, if it exists. - Amiram Eldar, Feb 24 2024

Examples

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

Links

• Index entries for sequences related to powerful numbers.

Crossrefs

Intersection of A001694 and A059590.

Cf. A025494, A115644, A115646, A089359.

Programs

Mathematica

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
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 *)