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A038500
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Highest power of 3 dividing n.
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63
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1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 81
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OFFSET
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1,3
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COMMENTS
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To construct the sequence: start with 1 and concatenate twice: 1,1,1 then tripling the last term gives: 1,1,3. Concatenating those 3 terms twice gives: 1,1,3,1,1,3,1,1,3, triple the last term -> 1,1,3,1,1,3,1,1,9. Concatenating those 9 terms twice gives: 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9, triple the last term -> 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,27 etc. - Benoit Cloitre, Dec 17 2002
Also 3-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 3-adic absolute value. - Wolfdieter Lang, Jun 28 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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Multiplicative with a(p^e) = p^e if p = 3, 1 otherwise. - Mitch Harris, Apr 19 2005
a(n) = gcd(n,3^n).
O.g.f.: x/(1 - x) + 2*Sum_{n >= 1} 3^(n-1)*x^(3^n)/ (1 - x^(3^n)). (End)
Sum_{k=1..n} a(k) ~ (2/(3*log(3)))*n*log(n) + (2/3 + 2*(gamma-1)/(3*log(3)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022
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MAPLE
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MATHEMATICA
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Flatten[{1, 1, #}&/@(3^IntegerExponent[#, 3]&/@(3*Range[40]))] (* or *) hp3[n_]:=If[Divisible[n, 3], 3^IntegerExponent[n, 3], 1]; Array[hp3, 90] (* Harvey P. Dale, Mar 24 2012 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, 3^valuation(n, 3))};
(Haskell)
a038500 = f 1 where
f y x = if m == 0 then f (y * 3) x' else y where (x', m) = divMod x 3
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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