A387287 Primes in the order of their first appearance among the factors of the averages of twin prime pairs.

2, 3, 5, 7, 17, 23, 11, 19, 47, 13, 29, 103, 107, 137, 43, 59, 41, 71, 31, 67, 139, 283, 149, 313, 37, 347, 373, 397, 443, 113, 467, 271, 181, 281, 577, 593, 199, 157, 653, 131, 101, 89, 241, 83, 251, 379, 773, 787, 167, 109, 907, 163, 73, 1033, 53, 223, 1117

Offset: 1

Author

Tamas Sandor Nagy, Aug 25 2025

Keywords

• nonn
• easy
• new

Comments

Will every prime appear, so that this sequence is a permutation of the primes?

The answer is yes if A071256(n) exists for every n. - Robert Israel, Aug 25 2025

Examples

			a(1) = 2 because 2 appeared first as a prime factor of the average of a twin prime pair, namely of 4 = 2*2 = 2^2, the average of 3 and 5, the first twin prime pair.
a(2) = 3 because 3 appeared next as a prime factor of the average of a twin prime pair, here 6 = 2*3, of the twin primes 5 and 7.
a(3) = 5 because 5 appeared next as a prime factor of the average of a twin prime pair, this time of 30 = 2*3*5, between 29 and 30. The averages 12 and 18 are skipped as their factors, 2 and 3, already appeared.
a(5) = 17 following a(4) = 7, skipping the primes 11 and 13 in the order of appearances.
		

Crossrefs

Cf. A000040, A014574, A071256, A071407, A090530, A294731.

Programs

Maple

P:= select(isprime, {seq(i,i=3..10^4,2)}):
TPA:= map(`+`, P intersect map(`-`,P,2),1):
TPA:= sort(convert(TPA,list)):
R:= NULL: S:= {}:
for t in TPA do
  V:= numtheory:-factorset(t) minus S;
  if nops(V) > 1 then printf("t = %d: %a\n",t,V) fi;
  R:= R, op(sort(convert(V,list)));
  S:= S union V;
od:
R; # Robert Israel, Aug 25 2025

Mathematica

With[{m = Select[Prime[Range[1000]], PrimeQ[# + 2] &] + 1}, DeleteDuplicates[Flatten[FactorInteger[#][[;; , 1]] & /@ m]]] (* Amiram Eldar, Aug 25 2025 *)