A335961 Alternating factorions: Numbers m such that m = S_af(m) = af(d_1)+af(d_2)+...+af(d_k) where d_1 d_2 ... d_n is the decimal expansion of m and af(m) = m!-(m-1)!+(m-2)!+...1! (alternating factorial) with af(0) = 0 (base 10).

0, 1, 620, 621, 643

Offset: 1

Author

Andrzej Kukla, Jul 01 2020

Keywords

• nonn
• base
• fini
• full

Comments

Largest k such that S_af(k) > k is 1599999. That's why there are only five numbers such that S_af(m) = m. Proved by computer calculations.

If m has eight or more digits then S_af(m) < m. Proved directly.

Examples

			For m = 620, S_af(620) = af(6)+af(2)+af(0) = 619+1+0 = 620.
		

Crossrefs

Cf. A005165 (alternating factorial), A014080 (factorions).

Programs

Mathematica

af[0] = 0; af[n_] := af[n] = n! - af[n - 1]; Select[Range[1000], Total[af /@ IntegerDigits[#]] == # &] (* Amiram Eldar, Jul 02 2020 *)