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

Offset: 1

Author

Reinhard Zumkeller, Aug 27 2007

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

Cf. A001620, A082633.

Cf. A379003 (ordinal transform), A379004 (rgs-transform).

Cf. also A355582.

Programs

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

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 *)

PARI

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

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