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A343717
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a(n) is the smallest number that yields a prime when appended to n!.
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3
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1, 1, 3, 1, 1, 1, 7, 11, 29, 17, 43, 29, 13, 47, 19, 73, 37, 19, 41, 103, 41, 31, 43, 1, 113, 31, 37, 59, 41, 53, 41, 47, 1, 41, 149, 37, 53, 73, 337, 1, 103, 151, 293, 47, 107, 509, 127, 71, 167, 197, 167, 149, 67, 163, 139, 251, 59, 107, 241, 331, 269, 1, 149
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OFFSET
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0,3
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COMMENTS
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Appending to n! any number k <= n yields a multiple of k; that multiple cannot be prime except at k=1, so, for every n, a(n)=1 or a(n) > n.
a(n) = 1 iff n = 0 or n is in A024912.
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LINKS
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FORMULA
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EXAMPLE
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n=1: 1! = 1; appending a 1 yields 11, a prime, so a(1)=1.
n=2: 2! = 2; appending a 1 yields 21 = 3*7, and appending a 2 yields 22 = 2*11, but appending a 3 yields 23 (a prime), so a(2)=3.
n=19: 19! = 121645100408832000; appending any number < 103 yields a composite, but 121645100408832000103 is a prime, so a(19)=103.
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MAPLE
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a:= proc(n) option remember; local k, t; t:= n!;
for k while not isprime(parse(cat(t, k))) do od; k
end:
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MATHEMATICA
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Array[Block[{m = #!, k = 0}, While[! PrimeQ[10^If[k == 0, 1, IntegerLength[k]]*m + k], k++]; k] &, 62] (* Michael De Vlieger, May 17 2021 *)
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PROG
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(Python) # see link for faster program producing b-file
from sympy import factorial, isprime
def a(n):
start = str(factorial(n))
end = 1
while not isprime(int(start + str(end))): end += 2
return end
(PARI) for(n=0, 62, my(f=digits(n!)); forstep(k=1, oo, 2, my(p=fromdigits(concat(f, digits(k)))); if(ispseudoprime(p), print1(k, ", "); break))) \\ Hugo Pfoertner, May 18 2021
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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