A352834 -id:A352834 - 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.

1, 8, 3, 4, 5, 72, 7, 80, 108, 20, 11, 240, 13, 28, 15, 16, 17, 0, 19, 480, 21, 44, 23, 48, 25, 52, 27, 560, 29, 0, 31, 448, 33, 68, 35, 864, 37, 76, 39, 160, 41, 1680, 43, 880, 540, 92, 47, 144, 49, 200, 51, 1040, 53, 972, 55, 112, 57, 116, 59, 1920, 61, 124, 756, 64, 65

Offset: 1

Robert G. Wilson v, Feb 09 2004

nonn

k is a multiple of n.
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
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

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A090395, zeros are in A091896.
Cf. A000005 (number-of-divisors function).
Cf. A352834 (a(n)/n).

a = Table[0, {100}]; Do[b = Denominator[DivisorSigma[0, n]/n]; If[b < 101 && a[[b]] == 0, a[[b]] = n], {n, 1, 2640}]; a

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

a(n) = n*A352834(n). - M. F. Hasler, Apr 15 2022

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.

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