A372263 - OEIS

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

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OFFSET

1,1

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)

LINKS

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

FORMULA

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

EXAMPLE

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, ...

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

PROG

(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] */

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.

Sequence in context: A291358 A059650 A112244 * A210487 A071216 A069483

Adjacent sequences: A372260 A372261 A372262 * A372264 A372265 A372266

KEYWORD

nonn

AUTHOR

M. F. Hasler, Apr 24 2024

STATUS

approved