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

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

Offset: 1

Author

Vladimir Shevelev, Apr 24 2010, Apr 26 2010

Keywords

• nonn

Comments

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.

Examples

			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.
		

References

• Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature Sciences 4 (1996), 28-43.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponents, INTEGERS: El. J. of Combin. Number Theory, 7 (2007), Article #A33, 1-35.

Crossrefs

Cf. A050376, A025583, A164376.

Programs

Maple

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 16 2010

Mathematica

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 *)

Extensions

Edited and extended by R. J. Mathar, Jun 16 2010

More terms from Amiram Eldar, Oct 05 2023