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A290059
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a(n) = binomial(2*prime(n)-1, prime(n)-1) where prime(n) is the n-th prime.
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2
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3, 10, 126, 1716, 352716, 5200300, 1166803110, 17672631900, 4116715363800, 15033633249770520, 232714176627630544, 873065282167813104916, 212392290424395860814420, 3318776542511877736535400, 812850570172585125274307760, 3136262529306125724764953838760
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OFFSET
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1,1
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COMMENTS
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Charles Babbage (1791-1871) proved in 1819 that for every prime p > 2 this sequence is congruent to 1 (mod p^2).
Joseph Wolstenholme (1829-1891) proved in 1862 that for every prime p > 3 this sequence is congruent to 1 (mod p^3).
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LINKS
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FORMULA
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log(a(n)) ~ 2*log(2)*n * (log(n) + log(log(n)) - 1). - Vaclav Kotesovec, May 07 2022
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MAPLE
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seq(binomial(2*ithprime(i)-1, ithprime(i)-1), i=1..16);
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MATHEMATICA
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Array[Function[p, Binomial[2*p - 1, p - 1]]@ Prime@ # &, 16] (* Michael De Vlieger, Jul 19 2017 *)
Binomial[2#-1, #-1]&/@Prime[Range[20]] (* Harvey P. Dale, Mar 29 2024 *)
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PROG
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(PARI) a(n) = my(p=prime(n)); binomial(2*p-1, p-1); \\ Michel Marcus, Jul 19 2017
(Python)
from sympy import prime, binomial
def a(n):
p=prime(n)
return binomial(2*p - 1, p - 1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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