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A324511
Numbers m such that Product_{d|m} (sigma(d)/tau(d)) is an integer h where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).
1
1, 3, 5, 6, 7, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 25, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95
OFFSET
1,2
COMMENTS
Product_{d|n} (sigma(d)/tau(d)) >= 1 for all n >= 1.
Odd primes are terms.
Corresponding values of integers h: 1, 2, 3, 9, 4, 6, 98, 7, 36, 36, 9, 10, 64, 81, 12, 31, 784, 15, 6561, 16, ...
FORMULA
A324510(a(n)) = 1.
EXAMPLE
Product_{d|n} (sigma(d)/tau(d)) for n >= 1: 1, 3/2, 2, 7/2, 3, 9, 4, 105/8, 26/3, 81/4, 6, 98, 7, 36, 36, 651/8, ...
6 is a term because Product_{d|6} (sigma(d)/tau(d)) = sigma(1)/tau(1) * sigma(2)/tau(2) * sigma(3)/tau(3) * sigma(6)/tau(6) = 1/1 * 3/2 * 4/2 * 12/4 = 9 (integer).
MATHEMATICA
Select[Range[100], IntegerQ[Product[DivisorSigma[1, k]/DivisorSigma[0, k], {k, Divisors[#]}]] &] (* G. C. Greubel, Mar 04 2019 *)
PROG
(Magma) [n: n in [1..1000] | IsIntegral(&*[SumOfDivisors(d) / NumberOfDivisors(d): d in Divisors(n)])]
(Sage) [n for n in (1..100) if (product(sigma(k, 1)/sigma(k, 0) for k in n.divisors())).is_integer()] # G. C. Greubel, Mar 04 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Mar 03 2019
STATUS
approved