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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.
5
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
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
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Feb 09 2004
STATUS
approved