cp's OEIS Frontend

A091895 Least number k such that the denominator of d(k)/k = n, or zero if no such number exists, where d is the number-of-divisors function A000005.

%I A091895 #19 Apr 18 2022 10:33:52
%S A091895 1,8,3,4,5,72,7,80,108,20,11,240,13,28,15,16,17,0,19,480,21,44,23,48,
%T A091895 25,52,27,560,29,0,31,448,33,68,35,864,37,76,39,160,41,1680,43,880,
%U A091895 540,92,47,144,49,200,51,1040,53,972,55,112,57,116,59,1920,61,124,756,64,65
%N A091895 Least number k such that the denominator of d(k)/k = n, or zero if no such number exists, where d is the number-of-divisors function A000005.
%C A091895 k is a multiple of n.
%C A091895 A search limit of 2*n^2 (as suggested by _Hugo Pfoertner_ on the SeqFan list) appears to be sufficient: Up to n = 10^5, the largest ratio r(n) = a(n)/n is r(90090) = 672. - _M. F. Hasler_, Apr 04 2022
%C A091895 It appears that even a(n) <= 16*n^(4/3), verified up to n = 10^6 with search limit 2*n^2. Large values of a(n)/n^(4/3) are reached in particular at multiples of 2*3*5*7*11, but also at 2^3*3^3*5*11*13. See A352834 for more. - _M. F. Hasler_, Apr 15 2022
%H A091895 M. F. Hasler, <a href="/A091895/b091895.txt">Table of n, a(n) for n = 1..1000</a>
%F A091895 a(n) = n*A352834(n). - _M. F. Hasler_, Apr 15 2022
%t A091895 a = Table[0, {100}]; Do[b = Denominator[DivisorSigma[0, n]/n]; If[b < 101 && a[[b]] == 0, a[[b]] = n], {n, 1, 2640}]; a
%o A091895 (PARI) apply( {A091895(n,L=n^2*2)=forstep(k=n,L,n,denominator(numdiv(k)/k)==n&&return(k))}, [1..99]) \\ _M. F. Hasler_, Apr 04 2022
%Y A091895 Cf. A090395, zeros are in A091896.
%Y A091895 Cf. A000005 (number-of-divisors function).
%Y A091895 Cf. A352834 (a(n)/n).
%K A091895 nonn
%O A091895 1,2
%A A091895 _Robert G. Wilson v_, Feb 09 2004