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

%I A387287 #22 Aug 29 2025 00:18:55
%S A387287 2,3,5,7,17,23,11,19,47,13,29,103,107,137,43,59,41,71,31,67,139,283,
%T A387287 149,313,37,347,373,397,443,113,467,271,181,281,577,593,199,157,653,
%U A387287 131,101,89,241,83,251,379,773,787,167,109,907,163,73,1033,53,223,1117
%N A387287 Primes in the order of their first appearance among the factors of the averages of twin prime pairs.
%C A387287 Will every prime appear, so that this sequence is a permutation of the primes?
%C A387287 The answer is yes if A071256(n) exists for every n. - _Robert Israel_, Aug 25 2025
%e A387287 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.
%e A387287 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.
%e A387287 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.
%e A387287 a(5) = 17 following a(4) = 7, skipping the primes 11 and 13 in the order of appearances.
%p A387287 P:= select(isprime, {seq(i,i=3..10^4,2)}):
%p A387287 TPA:= map(`+`, P intersect map(`-`,P,2),1):
%p A387287 TPA:= sort(convert(TPA,list)):
%p A387287 R:= NULL: S:= {}:
%p A387287 for t in TPA do
%p A387287   V:= numtheory:-factorset(t) minus S;
%p A387287   if nops(V) > 1 then printf("t = %d: %a\n",t,V) fi;
%p A387287   R:= R, op(sort(convert(V,list)));
%p A387287   S:= S union V;
%p A387287 od:
%p A387287 R; # _Robert Israel_, Aug 25 2025
%t A387287 With[{m = Select[Prime[Range[1000]], PrimeQ[# + 2] &] + 1}, DeleteDuplicates[Flatten[FactorInteger[#][[;; , 1]] & /@ m]]] (* _Amiram Eldar_, Aug 25 2025 *)
%Y A387287 Cf. A000040, A014574, A071256, A071407, A090530, A294731.
%K A387287 nonn,easy,new
%O A387287 1,1
%A A387287 _Tamas Sandor Nagy_, Aug 25 2025