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A176297
Numbers with at least one 3 in their prime signature.
11
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved