A386809 Numbers that have exactly two exponents in their prime factorization that are equal to 5.

7776, 38880, 54432, 85536, 100000, 101088, 132192, 147744, 178848, 194400, 225504, 241056, 272160, 287712, 300000, 318816, 334368, 365472, 381024, 412128, 427680, 458784, 474336, 505440, 520992, 537824, 552096, 567648, 598752, 614304, 645408, 660960, 692064, 700000

Offset

1,1

Comments

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^5 + 1/p^6) * ((Sum_{p prime} (p-1)/(p^6 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^6 - p + 1)^2)) / 2 = 4.86539910559896710587...*10^(-5) (Elma and Martin, 2024).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.

Index entries for sequences computed from indices in prime factorization.

Mathematica

f[p_, e_] := If[e == 5, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[700000], s[#] == 2 &]

Prog

(PARI) isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 2;

Crossrefs

Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), A386801 (k=3), A386805 (k=4), this sequence (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 5: A386807 (m=0), A386808 (m=1), this sequence (m=2), A386810 (m=3).

Sequence in context: A176375 A210116 A223509 * A187355 A190465 A064782

Adjacent sequences: A386806 A386807 A386808 * A386810 A386811 A386812

Keyword

nonn,easy

Author

Amiram Eldar, Aug 03 2025

Status

approved