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A263882
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Babbage quotients b_p = (binomial(2p-1, p-1) - 1)/p^2 with p = prime(n).
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4
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1, 5, 35, 2915, 30771, 4037381, 48954659, 7782070631, 17875901604959, 242158352370063, 637739431824553035, 126348774791431208099, 1794903484322270273951, 367972191114796344623951, 1116504994413003106003899551, 3498520498083111051973370669639
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OFFSET
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2,2
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COMMENTS
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Charles Babbage proved in 1819 that b_p is an integer for prime p > 2. In 1862 Wolstenholme proved that the Wolstenholme quotient W_p = b_p / p is an integer for prime p > 3; see A034602.
The quotient b_n is an integer for composite n in A267824. No composite n is known for which W_n is an integer.
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.
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LINKS
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FORMULA
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a(n) = prime(n)*A034602(n) for n > 2.
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EXAMPLE
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a(2) = (binomial(2*3-1,3-1) - 1)/3^2 = (binomial(5,2) - 1)/9 = (10-1)/9 = 1.
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MAPLE
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map(p -> (binomial(2*p-1, p-1)-1)/p^2, select(isprime, [seq(i, i=3..100, 2)])); # Robert Israel, Nov 24 2015
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MATHEMATICA
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Table[(Binomial[2*Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^2, {n, 2, 17}]
Table[(Binomial[2p-1, p-1]-1)/p^2, {p, Prime[Range[2, 20]]}] (* Harvey P. Dale, Jul 20 2019 *)
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PROG
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(Magma) [(Binomial(2*NthPrime(n)-1, NthPrime(n)-1)-1)/NthPrime(n)^2: n in [2..20]]; // Vincenzo Librandi, Nov 25 2015
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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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