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A176699
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Fermi-Dirac composite numbers that are not a sum of two Fermi-Dirac primes (A050376).
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4
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145, 187, 205, 217, 219, 221, 247, 301, 325, 343, 415, 427, 475, 517, 535, 553, 555, 583, 637, 667, 671, 697, 715, 781, 783, 793, 795, 805, 807, 817, 835, 847, 851, 871, 895, 901, 905, 925, 959, 1003, 1005, 1027, 1045, 1057, 1059, 1075, 1081, 1135, 1141, 1147
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OFFSET
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1,1
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COMMENTS
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We define a Fermi-Dirac composite number as a positive integer with at least two factors in its factorization over distinct terms of A050376.
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.
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REFERENCES
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Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature Sciences 4 (1996), 28-43.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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:
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:
for n from 2 to 1000 do if isA176699(n) then printf("%d, \n", n) ; end if; end do: # R. J. Mathar, Jun 160 2010
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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