

A108911


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


5



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
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OFFSET

1,3


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 nonpositional 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


LINKS

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


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


EXAMPLE

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


MAPLE

seq(nadd((floor(n/(10^j))10*floor(n/(10^(j+1))))!, j=0..ilog10(n)), n=0..53);
# second Maple program:
a:= n> nadd(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 *)


PROG

(PARI) a(n) = my(d = digits(n)); n  sum(i=1, #d, d[i]!); \\ Michel Marcus, Apr 21 2014
(MAGMA) [n&+[Factorial(d): d in Intseq(n)]: n in [1..60]]; // Bruno Berselli, Oct 25 2018


CROSSREFS

Cf. A005096, A014080, A061602.
Sequence in context: A154641 A178711 A257067 * A005096 A275796 A164535
Adjacent sequences: A108908 A108909 A108910 * A108912 A108913 A108914


KEYWORD

sign,base,look,easy


AUTHOR

Paolo P. Lava and Giorgio Balzarotti, Jul 18 2005


EXTENSIONS

Extended by Ray Chandler, Jul 24 2005


STATUS

approved



