A386805 Numbers that have exactly two exponents in their prime factorization that are equal to 4.

1296, 6480, 9072, 10000, 14256, 16848, 22032, 24624, 29808, 30000, 32400, 37584, 38416, 40176, 45360, 47952, 50625, 53136, 55728, 60912, 63504, 68688, 70000, 71280, 76464, 79056, 84240, 86832, 90000, 92016, 94608, 99792, 101250, 102384, 107568, 110000, 110160, 115248

Offset

1,1

Comments

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^4 + 1/p^5) * ((Sum_{p prime} (p-1)/(p^5 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^5 - p + 1)^2)) / 2 = 0.00032582100547959312658... (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 == 4, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[120000], s[#] == 2 &]

Prog

(PARI) isok(k) = vecsum(apply(x -> if(x == 4, 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), this sequence (k=4), A386809 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 4: A386803 (m=0), A386804 (m=1), this sequence (m=2), A386806 (m=3).

Sequence in context: A250810 A378900 A378768 * A320893 A390127 A276282

Adjacent sequences: A386802 A386803 A386804 * A386806 A386807 A386808

Keyword

nonn,easy

Author

Amiram Eldar, Aug 03 2025

Status

approved