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A176297 Numbers with at least one 3 in their prime signature. 7
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 (list; graph; refs; listen; history; text; internal format)
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: A301517 A195086 A336593 * A175496 A048109 A068781

Adjacent sequences:  A176294 A176295 A176296 * A176298 A176299 A176300

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

STATUS

approved

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Last modified October 22 10:16 EDT 2021. Contains 348160 sequences. (Running on oeis4.)