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A176699 Fermi-Dirac composite numbers that are not a sum of two Fermi-Dirac primes (A050376).

%I A176699 #21 Jun 29 2025 21:40:43
%S A176699 145,187,205,217,219,221,247,301,325,343,415,427,475,517,535,553,555,
%T A176699 583,637,667,671,697,715,781,783,793,795,805,807,817,835,847,851,871,
%U A176699 895,901,905,925,959,1003,1005,1027,1045,1057,1059,1075,1081,1135,1141,1147
%N A176699 Fermi-Dirac composite numbers that are not a sum of two Fermi-Dirac primes (A050376).
%C A176699 We define a Fermi-Dirac composite number as a positive integer with at least two factors in its factorization over distinct terms of A050376.
%C A176699 They are those c for which A064547(c) >= 2, namely c= 6, 8, 10, 12,..., 62, 63, 64, 65, ..., or the complement of A050376 with respect to the natural numbers > 1.
%D A176699 Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature Sciences 4 (1996), 28-43.
%H A176699 Amiram Eldar, <a href="/A176699/b176699.txt">Table of n, a(n) for n = 1..10000</a>
%H A176699 Simon Litsyn and Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponents</a>, INTEGERS: El. J. of Combin. Number Theory, 7 (2007), Article #A33, 1-35.
%e A176699 291 = 3*97 is a Fermi-Dirac composite number, equal to 289+2, the sum of two Fermi-Dirac primes. Therefore 291 is not in the sequence.
%p A176699 A064547 := proc(n) f := ifactors(n)[2] ; a := 0 ; for p in f do a := a+wt(op(2, p)) ; end do: a ; end proc:
%p A176699 A050376 := proc(n) local a; if n = 1 then 2; else for a from procname(n-1)+1 do if A064547(a) = 1 then return a; end if; end do: end if; end proc:
%p A176699 isA176699 := proc(n) local pi,q ; if A064547(n) < 2 then return false; end if; for pi from 1 do if A050376(pi) > n then return true; else q := n-A050376(pi) ; if A064547(q) = 1 then return false; end if; end if; end do; end proc:
%p A176699 for n from 2 to 1000 do if isA176699(n) then printf("%d,\n",n) ; end if; end do: # R. J. Mathar, Jun 16 2010
%t A176699 pow2Q[n_] := n == 2^IntegerExponent[n, 2]; fdpQ[n_] := PrimePowerQ[n] && pow2Q[FactorInteger[n][[1, 2]]]; With[{m = 1200}, p = Select[Range[m], fdpQ]; Complement[Range[m], Join[{1}, p, Plus @@@ Subsets[p, {2}]]]] (* _Amiram Eldar_, Oct 05 2023 *)
%Y A176699 Cf. A050376, A025583, A164376.
%K A176699 nonn
%O A176699 1,1
%A A176699 _Vladimir Shevelev_, Apr 24 2010, Apr 26 2010
%E A176699 Edited and extended by _R. J. Mathar_, Jun 16 2010
%E A176699 More terms from _Amiram Eldar_, Oct 05 2023