A132741 - OEIS

A132741

Largest divisor of n having the form 2^i*5^j.

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 16, 1, 2, 1, 20, 1, 2, 1, 8, 25, 2, 1, 4, 1, 10, 1, 32, 1, 2, 5, 4, 1, 2, 1, 40, 1, 2, 1, 4, 5, 2, 1, 16, 1, 50, 1, 4, 1, 2, 5, 8, 1, 2, 1, 20, 1, 2, 1, 64, 5, 2, 1, 4, 1, 10, 1, 8, 1, 2, 25, 4, 1, 2, 1, 80, 1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 1, 2, 5, 32, 1, 2

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OFFSET

1,2

COMMENTS

The range of this sequence, { a(n); n>=0 }, is equal to A003592. - M. F. Hasler, Dec 28 2015

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = n / A132740(n).

a(A003592(n)) = A003592(n).

A051626(a(n)) = 0.

A007732(a(n)) = 1.

From R. J. Mathar, Sep 06 2011: (Start)

Multiplicative with a(2^e)=2^e, a(5^e)=5^e and a(p^e)=1 for p=3 or p>=7.

Dirichlet g.f. zeta(s)*(2^s-1)*(5^s-1)/((2^s-2)*(5^s-5)). (End)

a(n) = A006519(n)*A060904(n) = 2^A007814(n)*5^A112765(n). - M. F. Hasler, Dec 28 2015

Sum_{k=1..n} a(k) ~ n*(12*log(n)^2 + (24*gamma + 36*log(2) - 24)*log(n) + 24 - 24*gamma - 36*log(2) + 36*gamma*log(2) + 2*log(2)^2 - 18*log(5) + 18*gamma*log(5) + 27*log(2)*log(5) + 2*log(5)^2 + 18*log(5)*log(n) - 24*gamma_1)/(60*log(2)*log(5)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Jan 26 2023

MAPLE

A132741 := proc(n) local f, a; f := ifactors(n)[2] ; a := 1; for f in ifactors(n)[2] do if op(1, f) =2 then a := a*2^op(2, f) ; elif op(1, f) =5 then a := a*5^op(2, f) ; end if; end do; a; end proc: # R. J. Mathar, Sep 06 2011

MATHEMATICA

a[n_] := SelectFirst[Reverse[Divisors[n]], MatchQ[FactorInteger[#], {{1, 1}} | {{2, _}} | {{5, _}} | {{2, _}, {5, _}}]&]; Array[a, 100] (* Jean-François Alcover, Feb 02 2018 *)

a[n_] := Times @@ ({2, 5}^IntegerExponent[n, {2, 5}]); Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

PROG

(Haskell)

a132741 = f 2 1 where

f p y x | r == 0 = f p (y * p) x'

| otherwise = if p == 2 then f 5 y x else y

where (x', r) = divMod x p

-- Reinhard Zumkeller, Nov 19 2015

(PARI) A132741(n)=5^valuation(n, 5)<<valuation(n, 2) \\ M. F. Hasler, Dec 28 2015

CROSSREFS

Cf. A003592, A006519, A007732, A007814, A051626, A060904, A112765, A132740.

Cf. A001620, A082633.