A384020 Numbers k > 0 such that sigma(A018804(k)) = k*tau(A018804(k)) where sigma denotes the sum of divisors (A000203) and tau denotes the number of divisors (A000005).

1, 2, 3, 6, 7, 10, 14, 19, 21, 30, 31, 37, 38, 39, 42, 57, 62, 70, 74, 78, 79, 93, 97, 111, 114, 133, 139, 157, 158, 186, 190, 194, 199, 210, 211, 217, 222, 229, 237, 259, 266, 271, 273, 278, 291, 307, 310, 314, 331, 337, 367, 370, 379, 390, 398, 399, 410

Offset

1,2

Comments

Experimental result: the fraction of numbers k such that S(P(k)) > k*D(P(k)) tends to 0, S(P(k)) = k*D(P(k)) tends to 0, S(P(k)) < k*D(P(k)) tends to 1 with growing k where S(P(k)) denotes A000203(A018804(k)) and D(P(k)) denotes A000005(A018804(k)).

Links

Table of n, a(n) for n=1..57.

Example

For k = 6, a(6) = A000203(A018804(6)) = 6*A000005(A018804(6)).

Mathematica

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[500], Divide @@ DivisorSigma[{1, 0}, pil[#]] == # &] (* Amiram Eldar, May 17 2025 *)

Crossrefs

Cf. A000005, A000203, A018804, A382872.

Sequence in context: A351554 A002038 A032501 * A240212 A333551 A093677

Adjacent sequences: A384017 A384018 A384019 * A384021 A384022 A384023

Keyword

nonn

Author

Ctibor O. Zizka, May 17 2025

Status

approved