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

Author

Amiram Eldar, Aug 03 2025

Keywords

• nonn
• easy

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.

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).

Programs

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]

PARI

isok(k) = vecsum(apply(x -> if(x == 5, 1, 0), factor(k)[, 2])) == 3;