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A108911
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Difference between n and the sum of the factorials of its digits.
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5
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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
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = n - (N0! + N1! + N2! + ...) if n = N0*10^0 + N1*10^1 + N2*10^2 ...
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EXAMPLE
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For n = 35, a(35) = -91 because 35 - (3! + 5!) = 35 - (6 + 120) = -91.
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MAPLE
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seq(n-add((floor(n/(10^j))-10*floor(n/(10^(j+1))))!, j=0..ilog10(n)), n=0..53);
# second Maple program:
a:= n-> n-add(i!, i=convert(n, base, 10)):
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MATHEMATICA
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f[n_] := n - Plus @@ Factorial /@ IntegerDigits[n]; Table[f[n], {n, 53}] (* Ray Chandler, Jul 24 2005 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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