A386810 Numbers that have exactly three exponents in their prime factorization that are equal to 5.

24300000, 130691232, 170100000, 267300000, 315900000, 413100000, 461700000, 558900000, 653456160, 704700000, 753300000, 899100000, 996300000, 1044900000, 1142100000, 1190700000, 1252332576, 1287900000, 1433700000, 1437603552, 1482300000, 1628100000, 1680700000

Offset

1,1

Comments

The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^5 + 1/p^6) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 1.38560245036673575581*10^(-8), where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^6/(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.

Maple

M:= 10^10: # for terms <= M
B:= select(t -> ifactors(t)[2][.., 2]=[1, 1, 1], [$1..floor(M^(1/5))]):
R:= NULL:
for i from 1 to nops(B) do
  Q:= select(t -> igcd(t, B[i]) = 1 and not member(5, ifactors(t)[2][.., 2]), [$1 .. M/B[i]^5]);
  R:= R, op(B[i]^5 * Q);
od:
sort([R]); # Robert Israel, Aug 03 2025

Mathematica

seq[lim_] := Module[{s = {}, sqfs = Select[Range[Surd[lim, 5]], SquareFreeQ[#] && PrimeNu[#] == 3 &]}, Do[s = Join[s, sqf^5 * Select[Range[lim/sqf^5], CoprimeQ[#, sqf] && !MemberQ[FactorInteger[#][[;; , 2]], 5] &]], {sqf, sqfs}]; Union[s]]; seq[2*10^9]

Prog

(PARI) isok(k) = vecsum(apply(x -> if(x == 5, 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), A386806 (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), A386809 (m=2), this sequence (m=3).

Sequence in context: A392928 A251248 A135954 * A203260 A268290 A183791

Adjacent sequences: A386807 A386808 A386809 * A386811 A386812 A386813

Keyword

nonn,easy

Author

Amiram Eldar, Aug 03 2025

Status

approved