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A147844
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Difference between the number of distinct prime divisors of (2*n)!/n!^2 and pi(2*n), where pi(x) is the prime counting function.
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1
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0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 4, 5, 5, 5, 5, 6, 4, 3, 5, 6, 5, 4, 5, 5, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 8, 9, 8, 8, 10, 10, 11, 10, 10, 9, 9, 9, 9, 9, 9, 9, 8, 9, 10, 11, 11, 10, 10, 10, 10, 11, 10, 10, 11, 10, 10, 11, 11, 12, 12, 11, 12, 12, 12, 13, 13
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OFFSET
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1,7
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COMMENTS
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The expression (2*n)!/n!^2 is taken from C(2*n+1,n+1) - C(2*n,n) = (2*n)!/(n!^2*(n/(n+1)) = sum(k=1,n,C(n,k)*C(n,k-1)). This was posed in the Yahoo Group MathForFun, see link.
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LINKS
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EXAMPLE
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(2*10)!/10!^2 = 184756 = 2*2*11*13*17*19 which has 5 distinct divisors. Pi(2*10) = 8. 8-5=3 = a(10).
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MATHEMATICA
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Table[PrimePi[2n]-PrimeNu[(2n)!/(n!)^2], {n, 100}] (* Harvey P. Dale, Oct 30 2021 *)
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PROG
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(PARI) g2(n) = for(x=1, n, ct=omega((2*x)!/x!^2); print1(primepi(2*x)-ct", "))
(Magma) [#PrimesUpTo(2*n) - #PrimeDivisors( Factorial(2*n) div Factorial(n)^2):n in [1..91]]; // Marius A. Burtea, Nov 16 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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