A108911 Difference between n and the sum of the factorials of its digits.

0, 0, -3, -20, -115, -714, -5033, -40312, -362871, 8, 9, 9, 6, -11, -106, -705, -5024, -40303, -362862, 17, 18, 18, 15, -2, -97, -696, -5015, -40294, -362853, 23, 24, 24, 21, 4, -91, -690, -5009, -40288, -362847, 15, 16, 16, 13, -4, -99, -698, -5017, -40296, -362855, -71, -70, -70, -73

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Jul 18 2005

Keywords

• sign
• base
• look
• easy

Comments

Null values are at n = 1, 2, 145, 40585 (A014080). Twin values are at n = 1, 2; 11, 12; 21, 22; ... 10*i + 1, 10*i + 2. Not in sequence: 7, 10, 14, ... Nice polar diagrams repeating themselves with normalized angle to 9! and radius = a(n).

The sequence can be seen as the difference between the natural numbers in the decimal system (n_dec = N0*(10^0) + N1*(10^1) + N2*(10^2)...) and their values in a non-positional number system based on the factorials of the digits (n_fact = N0*(N0 - 1)! + N1*(N1 - 1)! + N2*(N2 - 1)! ...). See also A111095. Note that a(np) - a(n) is congruent to 0 mod 9 if n and np are different for the permutation of the digits. Example (a(5971) - a(1957))/9 = 446. The property can be easily derived by remembering that np - n is congruent to 0 mod 9. - Giorgio Balzarotti, Oct 15 2005

Examples

			For n = 35, a(35) = -91 because 35 - (3! + 5!) = 35 - (6 + 120) = -91.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005096, A014080, A061602.

Programs

Magma

[n-&+[Factorial(d): d in Intseq(n)]: n in [1..60]]; // Bruno Berselli, Oct 25 2018

Maple

a:= n-> n-add(i!, i=convert(n, base, 10)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 24 2018

Mathematica

f[n_] := n - Plus @@ Factorial /@ IntegerDigits[n]; Table[f[n], {n, 53}] (* Ray Chandler, Jul 24 2005 *)

PARI

a(n) = my(d = digits(n)); n - sum(i=1, #d, d[i]!); \\ Michel Marcus, Apr 21 2014

Formula

a(n) = n - (N0! + N1! + N2! + ...) if n = N0*10^0 + N1*10^1 + N2*10^2 ...

a(n) = n - A061602(n). - Michel Marcus, Apr 21 2014

Extensions

Extended by Ray Chandler, Jul 24 2005