A132741 - OEIS

%I #39 Dec 19 2024 17:34:39

%S 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,

%T 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,

%U 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

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

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

%H Reinhard Zumkeller, <a href="/A132741/b132741.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(A003592(n)) = A003592(n).

%F A051626(a(n)) = 0.

%F A007732(a(n)) = 1.

%F From _R. J. Mathar_, Sep 06 2011: (Start)

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

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

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

%F 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

%p 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

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

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

%o (Haskell)

%o a132741 = f 2 1 where

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

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

%o            where (x', r) = divMod x p

%o -- _Reinhard Zumkeller_, Nov 19 2015

%o (PARI) A132741(n)=5^valuation(n,5)<<valuation(n,2) \\ _M. F. Hasler_, Dec 28 2015

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

%Y Cf. A001620, A082633.

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

%Y Cf. also A355582.

%K nonn,easy,mult

%O 1,2

%A _Reinhard Zumkeller_, Aug 27 2007