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A372263 Least odd prime factor of the n-th sum of two consecutive primes, A001043(n) = prime(n) + prime(n+1), or 2 if there is no odd prime factor.

%I A372263 #18 Dec 30 2024 02:17:37
%S A372263 5,2,3,3,3,3,3,3,13,3,17,3,3,3,5,7,3,2,3,3,19,3,43,3,3,3,3,3,3,3,3,67,
%T A372263 3,3,3,7,5,3,5,11,3,3,3,3,3,5,7,3,3,3,59,3,3,127,5,7,3,137,3,3,3,3,3,
%U A372263 3,3,3,167,3,3,3,89,3,5,47,3,193,3,3,3,3,3,3,3,109,3,223
%N A372263 Least odd prime factor of the n-th sum of two consecutive primes, A001043(n) = prime(n) + prime(n+1), or 2 if there is no odd prime factor.
%C A372263 Since the sum of any two primes > 2 is even, we rather consider odd prime factors.
%C A372263 Can it be proved or disproved that there are primes that occur only finitely many times (or never) in this sequence? If so, which is the smallest such prime?
%C A372263 From _Robert Israel_, Dec 29 2024: (Start)
%C A372263 Dickson's conjecture implies that every odd prime occurs infinitely many times in the sequence.
%C A372263 a(n) = 2 if and only if n = A000720(2^k) where k is in A226178. (End)
%H A372263 Robert Israel, <a href="/A372263/b372263.txt">Table of n, a(n) for n = 1..10000</a>
%F A372263 a(n) = max(A078701(A001043(n)), 2) = A020639(max(A000265(A001043(n)), 2)), where A000265(m) > 2 unless m is in A000079.
%e A372263 Sums of two consecutive primes are given as s(n) = A001043(n). The least odd prime factor (or 2 if there's no odd prime factor) of these terms is a(n):
%e A372263 n = 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14,  15,  16,  17,  18, ...
%e A372263 s = 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, ...
%e A372263 a = 5, 2,  3,  3,  3,  3,  3,  3, 13,  3, 17,  3,  3,  3,   5,   7,   3,   2, ...
%e A372263 Also, a(21) = spf(152) = 19; a(23) = spf(172) = 43; a(32) = spf(268) = 67, ...
%p A372263 f:= proc(n) subs(infinity=2,min(numtheory:-factorset(ithprime(n)+ithprime(n+1)) minus {2})) end proc:
%p A372263 map(f, [$1..100]); # _Robert Israel_, Dec 29 2024
%o A372263 (PARI) apply( {a(n) = max(A078701(A001043(n)), 2)}, [1..99])
%o A372263 /* a "self-contained" but less efficient definition:
%o A372263 a(n) = factor(max((n=prime(n)+prime(n+1))>>valuation(n,2),2))[1,1] */
%Y A372263 Cf. A001043 (sums of two consecutive primes),  A078701 (least odd prime divisor), A020639 (spf: least prime factor), A000265 (odd part of n), A000079 (powers of 2).
%Y A372263 Cf. A024677 (spf of A024675(n) = A001043(n)/2).
%Y A372263 Cf. A226178.
%K A372263 nonn
%O A372263 1,1
%A A372263 _M. F. Hasler_, Apr 24 2024