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

1,1

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

Links

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

Example

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

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]

Prog

(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

Crossrefs

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

Sequence in context: A195086 A366761 A336593 * A175496 A048109 A068781

Adjacent sequences: A176294 A176295 A176296 * A176298 A176299 A176300

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

Status

approved