A316228 Numbers whose Fermi-Dirac prime factorization sums to a Fermi-Dirac prime.

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 28, 29, 31, 34, 36, 37, 39, 40, 41, 43, 46, 47, 48, 49, 52, 53, 55, 56, 58, 59, 61, 63, 66, 67, 71, 73, 76, 79, 81, 82, 83, 88, 89, 90, 94, 97, 100, 101, 103, 104, 107, 108, 109, 112

Offset: 1

Author

Gus Wiseman, Jun 27 2018

Keywords

• nonn

Comments

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0. Every positive integer has a unique factorization into distinct Fermi-Dirac primes.

Examples

			Sequence of multiarrows in the form "number: sum <= factors" begins:
   2:  2 <= {2}
   3:  3 <= {3}
   4:  4 <= {4}
   5:  5 <= {5}
   6:  5 <= {2,3}
   7:  7 <= {7}
   9:  9 <= {9}
  10:  7 <= {2,5}
  11: 11 <= {11}
  12:  7 <= {3,4}
  13: 13 <= {13}
  14:  9 <= {2,7}
  16: 16 <= {16}
  17: 17 <= {17}
  18: 11 <= {2,9}
  19: 19 <= {19}
  20:  9 <= {4,5}
  22: 13 <= {2,11}
  23: 23 <= {23}
  24:  9 <= {2,3,4}
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A050376, A064547, A100118, A213925, A299757, A305829, A316202, A316210, A316211, A316220.

Programs

Mathematica

FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
Select[Range[2,200],Length[FDfactor[Total[FDfactor[#]]]]==1&]