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A147844
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Difference between the number of distinct prime divisors of (2n)!/n!^2 and Pi(2n).
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,7
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COMMENTS
| The expression (2n)!/n!^2 is taken from C(2n+1,n+1) - C(2n,n) =
(2n)!/(n!^2*(n/(n+1)) = sum(k=1,n,C(n,k)*C(n,k-1)). This was posed in the
Yahoo Group MathForFun. A recent thread is shown in the link.
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LINKS
| MathForFun, Binomial Identity
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FORMULA
| Pi(x): number of primes <= n or the prime counting function.
omega(x): number of distinct prime divisors of x.
C(n,k): number of combinations of n things taken k at a time = n!/((n-k)!k!).
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EXAMPLE
| (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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PROG
| (PARI) g2(n) = for(x=1, n, ct=omega((2*x)!/x!^2); print1(primepi(2*x)-ct", "))
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CROSSREFS
| Cf. A001791
Sequence in context: A006641 A115756 A067731 * A130634 A053735 A033667
Adjacent sequences: A147841 A147842 A147843 * A147845 A147846 A147847
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KEYWORD
| nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)hotmail.com), Nov 15 2008
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EXTENSIONS
| Corrected the link. - Cino Hilliard (hillcino368(AT)hotmail.com), Nov 18 2008
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