A367504 a(1) = 2; for n > 1, a(n) = a(n-1) + 2*gpf(a(n-1)), where gpf(k) = A006530(k) = greatest prime dividing k.

2, 6, 12, 18, 24, 30, 40, 50, 60, 70, 84, 98, 112, 126, 140, 154, 176, 198, 220, 242, 264, 286, 312, 338, 364, 390, 416, 442, 476, 510, 544, 578, 612, 646, 684, 722, 760, 798, 836, 874, 920, 966, 1012, 1058, 1104, 1150, 1196, 1242, 1288, 1334, 1392, 1450, 1508, 1566, 1624, 1682, 1740, 1798

Offset: 1

Author

Scott R. Shannon, Nov 21 2023

Keywords

• nonn

Comments

Conjecture: with the requirement that the prime factorization of each term is written so that the primes are ordered from smallest to largest, the sequence is the lexicographically earliest infinite sequence of distinct positive numbers such that gpf(a(n-1)) * lpf(a(n)) = |a(n) - a(n-1)|, where gpf(k) = A006530(k) = greatest prime factor of k and lpf(k) = A020639(k) = least prime factor of k. In this way the sequence is the ordered prime factorization version of the 'Commas sequence', A121805. One can show that, for such a sequence to be infinite, no odd number can appear. Although for many terms a lower even number can be chosen for the following term, which can lead to even lower numbers for further terms, it is conjectured all such choices will ultimately halt the sequence as a number is eventually reached for which no unused next number exists which follows the required rule for the difference between the terms. Therefore all terms must be larger than the previous, and the earliest such infinite sequence is the given sequence.

See A367465 for the sequence when the requirement that the primes in the factorization of each term must be in order is removed.

Examples

			a(7) = 40 as a(6) = 30 = 2*3*5, thus A006530(30) = 5 and a(7) = a(6) + 2*5 = 30 + 2*5 = 40.
		

Crossrefs

Cf. A027746, A006530, A020639, A367465, A121805, A123581, A180107, A036441, A076271.

See also A367624.

Programs

Mathematica

NestList[#+2FactorInteger[#][[-1,1]]&,2,100] (* Paolo Xausa, Dec 31 2023 *)

Formula

a(n) = 2*A123581(n). The exponent of 2 in a(2n) is 1+A367624(n). - N. J. A. Sloane, Dec 06 2023