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%I #25 Jul 12 2021 03:38:55
%S 0,1,2,3,4,5,6,7,8,9,36,86,15960
%N Integers k such that k = (d1)_c + (d2)_c + ... + (dc)_c, where (d)_c denotes the descending factorial of d, c is the length of k and di is the i-th digit of k in base 10.
%C The descending factorial (d)_c is defined as d*(d-1)*(d-2)*...*(d-c+1).
%e (8)_2 + (6)_2 = 8*7 + 6*5 = 56 + 30 = 86, therefore 86 is in the list.
%t q[n_] := Module[{dig = IntegerDigits[n], nd}, nd = Length[dig]; Sum[d!/(d - nd)!, {d, dig}] == n]; Select[Range[0, 16000], q] (* _Amiram Eldar_, Jun 18 2021 *)
%o (C++) #include <iostream>
%o #include <cmath>
%o using namespace std; unsigned long long ff(unsigned long long a, int b){unsigned long long s=1;for(int i=0; i<b; i++){s=s*(a-i);} return s;}int main(int argc, char** argv) {int k, a, p=0; for(unsigned long long n=0; n<=pow(10,9); n++){k=floor(log10(n))+1;a=n;for(int j=1; j<=k; j++){p+=ff(a%10,k);a/=10; }if(p==n){cout<<n<<", ";} p=0;}}
%Y Cf. A014080 (factorions), A068424 (descending factorials), A345406.
%K nonn,fini,full,base
%O 1,3
%A _Andrzej Kukla_, Jun 18 2021
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