A330606 Numbers k such that k*d(k) and sigma(k) are relatively prime, where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

1, 2, 4, 8, 9, 16, 25, 36, 64, 81, 100, 121, 128, 144, 225, 256, 289, 324, 400, 484, 512, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1250, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2809, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761

Offset: 1

Author

Amiram Eldar, Dec 20 2019

Keywords

• nonn

Comments

If p is prime and p == 2 (mod 3) then p^2 is in the sequence.

Let E(x) = #{n | a(n) <= x} be the number of terms of this sequence up to x. Kanold proved that there are two constants 0 < c1 < c2 and a positive number x_0 such that c1 < E(x)/sqrt(x/log(x)) < c2 for x > x_0. De Koninck and Kátai proved that there is a positive constant c such that E(x) = c * (1 + o(1)) * sqrt(x/log(x)).

Apparently most of the terms are squares or powers of 2. Terms that are not included 1250, 4802, 31250, 57122, ...

Numbers k such that A099377(k) = A038040(k) and A099378(k) = A000203(k). - Amiram Eldar, Nov 02 2021

References

• József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 75.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jean-Marie De Koninck and Imre Kátai, On an estimate of Kanold, Int. J. Math. Anal., Vol. 5, No. 8 (2007), pp. 1-12.
• Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl II, Mathematische Annalen, Vol. 134, No. 3 (1958), pp. 225-231.

Crossrefs

Cf. A000005, A000203, A038040, A099377, A099378, A324121.

Subsequence of A014567 and A046678.

Programs

Magma

[k:k in [1..5000]| Gcd(k*NumberOfDivisors(k),DivisorSigma(1,k)) eq 1]; // Marius A. Burtea, Dec 20 2019

Mathematica

Select[Range[10^4], CoprimeQ[# * DivisorSigma[0, #], DivisorSigma[1, #]] &]