A386806 Numbers that have exactly three exponents in their prime factorization that are equal to 4.

810000, 3111696, 5670000, 8910000, 10530000, 13770000, 15390000, 15558480, 18630000, 18974736, 23490000, 24010000, 25110000, 29970000, 33210000, 34228656, 34830000, 37015056, 38070000, 39690000, 40452048, 42930000, 47790000, 49410000, 52898832, 54270000, 57510000

Offset

1,1

Comments

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^4 + 1/p^5) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 4.77477224068657540815...*10^(-7), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^5/(p-1)-1))^m (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

seq[lim_] := Module[{s = {}, sqfs = Select[Range[Surd[lim, 4]], SquareFreeQ[#] && PrimeNu[#] == 3 &]}, Do[s = Join[s, sqf^4 * Select[Range[lim/sqf^4], CoprimeQ[#, sqf] && !MemberQ[FactorInteger[#][[;; , 2]], 4] &]], {sqf, sqfs}]; Union[s]]; seq[6*10^7]

Prog

(PARI) isok(k) = vecsum(apply(x -> if(x == 4, 1, 0), factor(k)[, 2])) == 3;

Crossrefs

Numbers that have exactly three exponents in their prime factorization that are equal to k: A386798 (k=2), A386802 (k=3), this sequence (k=4), A386810 (k=5).

Numbers that have exactly m exponents in their prime factorization that are equal to 4: A386803 (m=0), A386804 (m=1), A386805 (m=2), this sequence (m=3).

Sequence in context: A270117 A178290 A083617 * A343082 A344239 A344240

Adjacent sequences: A386803 A386804 A386805 * A386807 A386808 A386809

Keyword

nonn,easy

Author

Amiram Eldar, Aug 03 2025

Status

approved