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.

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, 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, 3, 3, 3, 167, 3, 3, 3, 89, 3, 5, 47, 3, 193, 3, 3, 3, 3, 3, 3, 3, 109, 3, 223

Offset: 1

Author

M. F. Hasler, Apr 24 2024

Keywords

• nonn

Comments

Since the sum of any two primes > 2 is even, we rather consider odd prime factors.

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?

From Robert Israel, Dec 29 2024: (Start)

Dickson's conjecture implies that every odd prime occurs infinitely many times in the sequence.

a(n) = 2 if and only if n = A000720(2^k) where k is in A226178. (End)

Examples

			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):
n = 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14,  15,  16,  17,  18, ...
s = 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, ...
a = 5, 2,  3,  3,  3,  3,  3,  3, 13,  3, 17,  3,  3,  3,   5,   7,   3,   2, ...
Also, a(21) = spf(152) = 19; a(23) = spf(172) = 43; a(32) = spf(268) = 67, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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).

Cf. A024677 (spf of A024675(n) = A001043(n)/2).

Cf. A226178.

Programs

Maple

f:= proc(n) subs(infinity=2,min(numtheory:-factorset(ithprime(n)+ithprime(n+1)) minus {2})) end proc:
map(f, [$1..100]); # Robert Israel, Dec 29 2024

PARI

apply( {a(n) = max(A078701(A001043(n)), 2)}, [1..99])
/* a "self-contained" but less efficient definition:
a(n) = factor(max((n=prime(n)+prime(n+1))>>valuation(n,2),2))[1,1] */

Formula

a(n) = max(A078701(A001043(n)), 2) = A020639(max(A000265(A001043(n)), 2)), where A000265(m) > 2 unless m is in A000079.