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A060904
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Largest power of 5 that divides n.
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17
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1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1
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OFFSET
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1,5
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COMMENTS
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Also the largest power of 5 that divides the n-th Fibonacci number A000045(n).
Multiplicative with a(p^e) = 5^e if p = 5, else a(p^e) = 1. - Mitch Harris, Apr 19 2005
Also 5-adic value of 1/n, n >= 1. See the Mahler reference, definition on p. 7. This is a non-archimedean valuation. See Mahler, p. 10. Sometimes also called 5-adic absolute value. - Wolfdieter Lang, Jun 30 2014
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REFERENCES
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Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.
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LINKS
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Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
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FORMULA
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If n is not divisible by 5, then a(n) = 1. If n = 5^k * m where m is not divisible by 5, then a(n) = 5^k.
Dirichlet g.f.: zeta(s)*(5^s-1)/(5^s-5). - R. J. Mathar, Jul 12 2012
a(n) = gcd(n,5^n).
O.g.f.: x/(1 - x) + 4*Sum_{n >= 1} 5^(n-1)*x^(5^n)/ (1 - x^(5^n)). (End).
a(n) = (1/5)*(sigma(5*n) - sigma(n))/(sigma(5*n) - 5*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
Sum_{k=1..n} a(k) ~ (4/(5*log(5)))*n*log(n) + (3/5 + 4*(gamma-1)/(5*log(5)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022
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EXAMPLE
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a(10) = 5 because 10 = 5 * 2.
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MAPLE
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MATHEMATICA
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PROG
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(Sage) [5^valuation(i, 5) for i in [1..100]] # Tom Edgar, Mar 22 2014
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
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STATUS
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approved
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