A047727 -id:A047727 - cp's OEIS frontend

A069810 Integers k such that gcd(k, sigma(k)) = tau(k).

1, 56, 60, 96, 132, 184, 204, 248, 276, 348, 376, 480, 492, 504, 564, 568, 612, 632, 636, 708, 824, 852, 996, 1016, 1068, 1208, 1212, 1248, 1284, 1336, 1356, 1528, 1572, 1592, 1632, 1644, 1784, 1788, 1908, 1912, 1980, 2004, 2076, 2104, 2148, 2168, 2232

Offset: 1

Benoit Cloitre, Apr 30 2002

			The divisors of 56 are 1, 2, 4, 7, 8, 14, 28, 56, so tau(56) = 8 and sigma(56) = 120. As gcd(56, 120) = tau(56) = 8, so 56 belongs to this sequence. - _Bernard Schott_, Oct 18 2019

Giovanni Resta, Table of n, a(n) for n = 1..10000

Subsequence of A047727 (numbers that are arithmetic and refactorable).

Cf. A003601 (arithmetic numbers), A033950 (refactorable numbers).

[k:k in [1..2300]| Gcd(k,DivisorSigma(1,k)) eq #Divisors(k)]; // Marius A. Burtea, Oct 18 2019

Select[Range[2500], GCD[DivisorSigma[1, #], #] == DivisorSigma[0, #] &] (* Jayanta Basu, Mar 21 2013 *)

for(n=1,3000,if(gcd(n, sigma(n))==numdiv(n),print1(n,",")))

A277368 Numbers such that the number of their divisors divide the sum of their aliquot parts.

Original entry on oeis.org

1, 4, 10, 16, 25, 26, 34, 56, 58, 60, 64, 74, 81, 82, 90, 96, 100, 106, 120, 121, 122, 132, 146, 178, 184, 194, 202, 204, 216, 218, 226, 234, 248, 274, 276, 289, 298, 306, 312, 314, 346, 348, 362, 364, 376, 386, 394, 408, 440, 458, 466, 480, 482, 492, 504, 514

Offset: 1

Paolo P. Lava, Oct 11 2016

If p is a prime such that p == 2 (mod 3) then p^2 is a term. Bateman et al. (1981) proved that the asymptotic density of this sequence is 0. - Amiram Eldar, Jan 16 2020

			sigma(26) - 26 = 42 - 26 = 16, d(26) = 4 and 16 / 4 = 4.

Richard G. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, chapter 2, p. 76.

Paolo P. Lava, Table of n, a(n) for n = 1..1000
Paul T. Bateman, Paul Erdős, Carl Pomerance and E.G. Straus, The arithmetic mean of the divisors of an integer, in Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol. 899, Springer, Berlin - New York, 1981, pp. 197-220, alternative link.

Cf. A000005, A001065, A003601, A047727.

[k:k in [1..550]| (DivisorSigma(1,k)-k) mod DivisorSigma(0,k) eq 0]; // Marius A. Burtea, Jan 16 2020

with(numtheory): P:= proc(q) local n; for n from 1 to q do
if type((sigma(n)-n)/tau(n),integer) then print(n); fi; od; end: P(10^3);

Select[Range@ 520, Mod[DivisorSigma[1, #] - #, DivisorSigma[0, #]] == 0 &] (* Michael De Vlieger, Oct 14 2016 *)

isok(n) = ((sigma(n) - n) % numdiv(n)) == 0; \\ Michel Marcus, Oct 11 2016

Solutions k to A000005(k) | A001065(k).

A333638 Numbers m such that (m * sigma(m)) / tau(m) is an integer k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Jaroslav Krizek, Mar 30 2020

nonn

Corresponding values of integers k: 1, 3, 6, 15, 18, 28, 30, 39, 45, 66, 56, 91, 84, 90, 153, 117, 190, 140, 168, ...
Supersequence of refactorable (A033950), arithmetic (A003601) and refactorable arithmetic numbers (A047727).
Sequence of numbers from this sequence that are neither refactorable nor arithmetic: 10, 26, 32, 34, 50, 58, 63, 74, 75, 82, 90, 98, 106, 117, 120, 122, 130, 146, ...

			10 is a term because (10 * sigma(10)) / tau(10) = (10 * 18) / 4 = 45 (integer).

Cf. A000005, A000203, A033950, A047727.

[m: m in [1..10^5] | IsIntegral((&+Divisors(m) * m) / #Divisors(m))]

Select[Range[100], Divisible[# * DivisorSigma[1, #], DivisorSigma[0, #]] &] (* Amiram Eldar, Mar 31 2020 *)

isok(m) = (m*sigma(m) % numdiv(m)) == 0; \\ Michel Marcus, Mar 31 2020

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A069810 Integers k such that gcd(k, sigma(k)) = tau(k).

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A277368 Numbers such that the number of their divisors divide the sum of their aliquot parts.

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A333638 Numbers m such that (m * sigma(m)) / tau(m) is an integer k.

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