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

1,2

Comments

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

Links

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

Formula

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A090395, zeros are in A091896.

Cf. A000005 (number-of-divisors function).

Cf. A352834 (a(n)/n).

Sequence in context: A070474 A070597 A222232 * A111436 A014549 A346441

Adjacent sequences: A091892 A091893 A091894 * A091896 A091897 A091898

Keyword

nonn

Author

Robert G. Wilson v, Feb 09 2004

Status

approved