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A125503
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Smallest number k such that the numerator of the generalized harmonic number H(k,n) = Sum_{i=1..k} 1/i^n is a prime.
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1
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2, 2, 3, 2, 23, 73, 15, 2, 3, 5, 13, 57, 3, 171, 5, 2, 21, 7, 55, 8902, 26, 1298, 115, 139, 3, 2019, 3, 4, 3, 15, 56, 177
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OFFSET
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1,1
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COMMENTS
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a(n) = 2 for n = {1,2,4,8,16,...}. Corresponding Fermat primes A019434.
a(n) = 3 for n = {3,9,13,25,27,29,95,107,153,159,...}.
a(n) = 5 for n = {10,15,60,90,197,209,...}.
a(n) = 7 for n = {18,47,112,155,273,...}.
a(n) = 15 for n = {7,30,43,...}.
a(21) = 26. a(28) = 4. a(31) = 56. a(144) = 9.
a(22)-a(25) = {1298,115,139,3}.
a(27)-a(32) = {3,4,3,15,56,177}.
a(n) = 3 for all n>2 listed in A125706. (End)
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LINKS
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MATHEMATICA
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Do[n = 1; f = 0; While[Not[PrimeQ[Numerator[f]]], f = f + 1/n^x; n++ ]; Print[{x, n - 1}], {x, 1, 25}] (* Alexander Adamchuk, Apr 18 2010 *)
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PROG
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(PARI) a(n) = my(k=1); while (!ispseudoprime(numerator(sum(i=1, k, 1/i^n))), k++); k; \\ Michel Marcus, Jun 04 2022
(Python)
from sympy import isprime
from fractions import Fraction
def a(n):
Hkn, k = Fraction(1, 1), 1
while not isprime(Hkn.numerator):
k += 1
Hkn += Fraction(1, k**n)
return k
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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