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A387216 Numbers that have at least two prime factors (counting multiplicity) congruent to 1 mod 3.

%I A387216 #20 Aug 26 2025 09:51:38
%S A387216 49,91,98,133,147,169,182,196,217,245,247,259,266,273,294,301,338,343,
%T A387216 361,364,392,399,403,427,434,441,455,469,481,490,494,507,511,518,532,
%U A387216 539,546,553,559,588,589,602,637,651,665,676,679,686,703,721,722,728,735,741,763,777,784,793,798
%N A387216 Numbers that have at least two prime factors (counting multiplicity) congruent to 1 mod 3.
%H A387216 Vincenzo Librandi, <a href="/A387216/b387216.txt">Table of n, a(n) for n = 1..7923</a>
%H A387216 Hajrudin Fejzić, <a href="https://arxiv.org/abs/2508.14937">Nontrivial Solutions to a Cubic Identity and the Factorization of n^2+n+1</a>, arXiv:2508.14937 [math.GM], 2025. See Theorem 2 p. 4.
%t A387216 ff[{m_,n_}]:=Table[m,n];Select[Range[798],Count[Mod[ff/@FactorInteger[#]//Flatten,3],1]>1&] (* _James C. McMahon_, Aug 22 2025 *)
%o A387216 (PARI) isok(k) = my(f=factor(k)); sum(i=1, #f~, if ((f[i,1]%3) == 1, f[i,2])) >= 2;
%o A387216 (Magma) res := [];for n in [1..1000] do L := [ f[2] : f in Factorization(n) | f[1] mod 3 eq 1 ]; count := (#L eq 0) select 0 else &+L;if count gt 1 then Append(~res, n); end if; end for; res;
%o A387216  // _Vincenzo Librandi_, Aug 24 2025
%Y A387216 Cf. A002476 (primes congruent to 1 mod 3), A050931 (at least one).
%K A387216 nonn,new
%O A387216 1,1
%A A387216 _Michel Marcus_, Aug 22 2025