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

%I A176297 #33 Aug 24 2021 06:15:35
%S A176297 8,24,27,40,54,56,72,88,104,108,120,125,135,136,152,168,184,189,200,
%T A176297 216,232,248,250,264,270,280,296,297,312,328,343,344,351,360,375,376,
%U A176297 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
%N A176297 Numbers with at least one 3 in their prime signature.
%C A176297 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.
%C A176297 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
%H A176297 Robert Israel, <a href="/A176297/b176297.txt">Table of n, a(n) for n = 1..10000</a>
%e A176297 8=2^3, 24=2^3*3, 27=3^3, 40=2^3*5, ...
%p A176297 filter:= proc(x) local F; F:= map(t->t[2],ifactors(x)[2]);has(F,3) end proc:
%p A176297 select(filter, [$1..1000]); # _Robert Israel_, Jan 11 2015
%p A176297 # alternative:
%p A176297 isA176297 := proc(n)
%p A176297     local p;
%p A176297     for p in ifactors(n)[2] do
%p A176297         if op(2,p) = 3 then
%p A176297             return true;
%p A176297         end if;
%p A176297     end do:
%p A176297     false ;
%p A176297 end proc: # _R. J. Mathar_, Dec 08 2015
%t A176297 f[n_]:=MemberQ[Last/@FactorInteger[n],3]; Select[Range[6!],f]
%o A176297 (PARI) isok(n) = vecsearch(vecsort(factor(n)[,2]), 3); \\ _Michel Marcus_, Jan 11 2015
%o A176297 (Python)
%o A176297 from sympy import factorint
%o A176297 def ok(n): return 3 in [e for e in factorint(n).values()]
%o A176297 print(list(filter(ok, range(713)))) # _Michael S. Branicky_, Aug 24 2021
%Y A176297 Cf. A000578, A001235, A176313, A176350, A109399, A176359.
%K A176297 nonn
%O A176297 1,1
%A A176297 _Vladimir Joseph Stephan Orlovsky_, Dec 07 2010