A109399 -id:A109399 - 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

A386801 Numbers that have exactly two exponents in their prime factorization that are equal to 3.

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

Offset: 1

Amiram Eldar, Aug 03 2025

Subsequence of A109399 and first differs from it at n = 64: A109399(64) = 27000 = 2^3 * 3^3 * 5^3 is not a term of this sequence.
Numbers k such that A295883(k) = 2.
The asymptotic density of this sequence is Product_{p primes} (1 - 1/p^3 + 1/p^4) * ((Sum_{p prime} (p-1)/(p^4 - p + 1))^2 - Sum_{p prime} ((p-1)^2/(p^4 - p + 1)^2)) / 2 = 0.0024403883082851652103... (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 A109399.
Cf. A295883.
Numbers that have exactly two exponents in their prime factorization that are equal to k: A386797 (k=2), this sequence (k=3), A386805 (k=4), A386809 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 3: A386799 (m=0), A386800 (m=1), this sequence (m=2), A386802 (m=3).

f[p_, e_] := If[e == 3, 1, 0]; s[1] = 0; s[n_] := Plus @@ f @@@ FactorInteger[n]; Select[Range[17000], s[#] == 2 &]

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

A176359 Numbers with at least three 3s in their prime signature.

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, 1704024, 1809000, 1852200, 1917000, 1971000, 2012472, 2079000, 2133000, 2148552

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

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 three values of i.

The asymptotic density of this sequence is 1 - (1 + s(1) + s(1)^2/2 - s(2)/2) * Product_{p prime} (1-1/p^3+1/p^4) = 0.000018992895371889141564..., where s(k) = Sum_{p prime} ((p-1)/(p^4-p+1))^k. - Amiram Eldar, Jul 22 2024

			27000 is a term since 27000 = 2^3 * 3^3 * 5^3.
74088 is a term since 74088 = 2^3 * 5^3 * 7^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Subsequence of A109399.

f[n_]:=Count[Last/@FactorInteger[n],3]>2; Select[Range[10!],f]

is(n) = #select(x -> x == 3, factor(n)[, 2]) > 2; \\ Amiram Eldar, Jul 22 2024

Edited by Matthew Vandermast, Dec 09 2010

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

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A386801 Numbers that have exactly two exponents in their prime factorization that are equal to 3.

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A176359 Numbers with at least three 3s in their prime signature.

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