A384276 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that is coprime to a(n-1) while the total number of prime factors, counted with multiplicity, of the form 4*k+1 and 4*k+3 for all terms a(1)..a(n) never differs by more than 1.

1, 2, 3, 4, 5, 6, 13, 7, 8, 15, 16, 17, 10, 9, 20, 11, 25, 12, 19, 26, 23, 29, 14, 37, 22, 35, 32, 39, 34, 31, 30, 41, 24, 53, 28, 51, 40, 43, 50, 21, 52, 45, 58, 47, 55, 61, 38, 65, 18, 73, 44, 75, 46, 85, 33, 64, 87, 68, 59, 60, 89, 48, 91, 74, 67, 70

Offset: 1

Author

Scott R. Shannon, May 24 2025

Keywords

• nonn

Comments

The terms are concentrated along four lines, although a closer examination shows both the top and bottom lines are composed of two separate lines that are entwined. The prime terms, which do not occur in their natural order, do not appear in the second-bottom line.

In the first 100000 terms the fixed points are 1, 2, 3, 4, 5, 6, 19, 59, 60, 4818, 4822, 7936, 8036, 8040, 9068, 9148, 10842; it is likely no more exist.

Examples

			a(6) = 6 as in a(1)..a(5) the total number of prime factors of the form 4*k+1 is one (5) while the total number of prime factors of the form 4*k+3 is one (3). As 6 only contains one prime factor of either form, and is coprime to 5, it can be chosen.
a(7) = 13 as in a(1)..a(6) the total number of prime factors of the form 4*k+1 is one (5) while the total number of prime factors of the form 4*k+3 is two (3,6). Therefore a(7) must contain between zero and two more prime factors of the form 4*k+1 than those of the form 4*k+3 while being coprime to 6. The smallest unused number meeting both of those conditions is 13.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 10000 terms. The colors are graduated across the spectrum to show the total number of prime factors of each term, with red being one prime factor. The green line is a(n) = n.
• Wikipedia, Chebyshev's bias.

Crossrefs

Cf. A382091, A381902, A007350, A038698, A027748.