A176359 -id:A176359 - cp's OEIS frontend

A176297 Numbers with at least one 3 in their prime signature.

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594, 600, 616, 621, 632, 648, 664, 675, 680, 686, 696, 702, 712

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

That is, if n = p1^e1 p2^e2 ... pr^er for distinct primes p1, p2,..., pr, then one of the exponents must be 3 for n to be in this sequence.

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^3 + 1/p^4) = 0.0952910730... - Amiram Eldar, Nov 14 2020

			8=2^3, 24=2^3*3, 27=3^3, 40=2^3*5, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000578, A001235, A176313, A176350, A109399, A176359.

filter:= proc(x) local F; F:= map(t->t[2],ifactors(x)[2]);has(F,3) end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 11 2015
# alternative:
isA176297 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) = 3 then
            return true;
        end if;
    end do:
    false ;
end proc: # R. J. Mathar, Dec 08 2015

f[n_]:=MemberQ[Last/@FactorInteger[n],3]; Select[Range[6!],f]

isok(n) = vecsearch(vecsort(factor(n)[,2]), 3); \\ Michel Marcus, Jan 11 2015

from sympy import factorint
def ok(n): return 3 in [e for e in factorint(n).values()]
print(list(filter(ok, range(713)))) # Michael S. Branicky, Aug 24 2021

A386802 Numbers that have exactly three exponents in their prime factorization that are equal to 3.

Original entry on oeis.org

27000, 74088, 189000, 287496, 297000, 343000, 351000, 370440, 459000, 474552, 513000, 621000, 783000, 814968, 837000, 963144, 999000, 1029000, 1061208, 1107000, 1157625, 1161000, 1259496, 1269000, 1323000, 1331000, 1407672, 1431000, 1437480, 1481544, 1593000, 1647000

Offset: 1

Amiram Eldar, Aug 03 2025

Subsequence of A176359 and first differs from it at n = 173: A176359(173) = 9261000 = 2^3 * 3^3 * 5^3 * 7^3 is not a term of this sequence.
Numbers k such that A295883(k) = 3.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^3 + 1/p^4) * (s(1)^3 + 3*s(1)*s(2) + 2*s(3)) / 6 = 0.000018940548516752487509..., where s(m) = (-1)^(m-1) * Sum_{p prime} (1/(p^4/(p-1)-1))^m (Elma and Martin, 2024).

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.

Subsequence of A176359.
Cf. A295883.
Numbers that have exactly three exponents in their prime factorization that are equal to k: A386798 (k=2), this sequence (k=3), A386806 (k=4), A386810 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), A386800 (m=1), A386801 (m=2), this sequence (m=3).

f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[2*10^6], s[#] == 3 &]

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

A109399 Numbers with at least two 3s in their prime signature.

Original entry on oeis.org

216, 1000, 1080, 1512, 2376, 2744, 2808, 3000, 3375, 3672, 4104, 4968, 5400, 6264, 6696, 6750, 7000, 7560, 7992, 8232, 8856, 9000, 9261, 9288, 10152, 10584, 10648, 11000, 11448, 11880, 12744, 13000, 13176, 13500, 13720, 14040, 14472, 15336, 15768, 16632, 17000, 17064, 17576, 17928, 18360, 18522, 19000, 19224, 19656, 20520, 20952, 21000, 21816, 22248, 23000, 23112, 23544, 23625, 24408, 24696, 24840, 25704, 26136, 27000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least two values of i.
Does not include all numbers with at least two unitary prime power divisors that are cubes (see example section).
The asymptotic density of this sequence is 1 - (1 + Sum_{p prime} ((p-1)/(p^4-p+1))) * Product_{p prime} (1-1/p^3+1/p^4) = 0.0024593812036570543518... . - Amiram Eldar, Jul 22 2024

			216 = 2^3*3^3, 1000 = 2^3*5^3, 1080 = 2^3*3^3*5, ...
On the other hand, 1728 = 2^6*3^3, 8000 = 2^6*5^3 and 21952 = 2^6*7^3 are not in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000578, A001235, A176297, A176313, A176350.

A176359 is a subsequence.

f[n_]:=Count[Last/@FactorInteger[n],3]>1; Select[Range[8!],f]

is(n)=#select(e->e==3, factor(n)[,2])>1 \\ Charles R Greathouse IV, Oct 19 2015

Edited by Matthew Vandermast, Dec 07 2010

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A176297 Numbers with at least one 3 in their prime signature.

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A386802 Numbers that have exactly three exponents in their prime factorization that are equal to 3.

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A109399 Numbers with at least two 3s in their prime signature.

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