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

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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

PARI

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

Python

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