A306639 Numbers m such that Sum_{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, 3, 5, 7, 10, 11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 33, 34, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 58, 59, 60, 61, 65, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 87, 89, 91, 93, 95, 97, 98, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113

Offset

1,2

Comments

Sum_{d|n} sigma(d)/tau(d) > 1 for all n > 1.

Sum_{d|n} sigma(d)/tau(d) = n only for numbers 1, 3, 10 and 30.

Odd primes are terms.

Corresponding values of integers h: 1, 3, 4, 5, 10, 7, 8, 12, 10, 11, 15, 13, 20, 16, 30, 17, 21, 25, 20, 20, 24, ...

Links

Table of n, a(n) for n=1..64.

Formula

A324500(a(n)) = 1.

Example

Sum_{d|n} sigma(d)/tau(d) for n >= 1: 1, 5/2, 3, 29/6, 4, 15/2, 5, 103/12, 22/3, 10, 7, 29/2, 8, 25/2, 12, 887/60, ...
10 is a term because Sum_{d|10} (sigma(d)/tau(d)) = sigma(1)/tau(1) + sigma(2)/tau(2) + sigma(5)/tau(5) + sigma(10)/tau(10) = 1/1 + 3/2 + 6/2 + 18/4 = 10 (integer).

Prog

(Magma) [n: n in [1..1000] | Denominator(&+[SumOfDivisors(d) / NumberOfDivisors(d): d in Divisors(n)]) eq 1]
(PARI) isok(n) = frac(sumdiv(n, d, sigma(d)/numdiv(d))) == 0; \\ Michel Marcus, Mar 03 2019

Crossrefs

Cf. A000005, A000203, A324499, A324500.

Sequence in context: A239929 A246955 A167907 * A190345 A189516 A138968

Adjacent sequences: A306636 A306637 A306638 * A306640 A306641 A306642

Keyword

nonn

Author

Jaroslav Krizek, Mar 02 2019

Status

approved