A382091 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1) while the total number of prime terms of the form 4*k + 1 is never less than those of the form 4*k + 3.

1, 2, 4, 6, 8, 10, 5, 15, 3, 9, 12, 14, 16, 18, 20, 22, 24, 21, 27, 30, 25, 35, 28, 26, 13, 39, 33, 11, 44, 32, 34, 17, 51, 36, 38, 19, 57, 42, 40, 45, 48, 46, 50, 52, 54, 56, 49, 63, 60, 55, 65, 70, 58, 29, 87, 66, 62, 31, 93, 69, 72, 64, 68, 74, 37, 111

Offset: 1

Author

Scott R. Shannon, Mar 15 2025

Keywords

• nonn

Comments

Unlike the EKG sequence A064413 the primes do not occur in their natural order, and the terms proceeding and following a prime can be more than two and three times respectively the value of the prime term. In the first 100000 terms the largest ratio found is a(75) = 84, a(76) = 7, with 84/7 = 12.

Like the EKG sequence the terms are predominantly concentrated along three lines, but with the addition of at least four more sparsely populated lines; see the attached image.

In the first 100000 terms the fixed points are 1, 2, 77, 121; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

Note that if the prime form selection rule is reversed so that the total number of primes of the form 4*k + 1 is never more than those of the form 4*k + 3 then the sequence will match the EKG sequence for the first 52082 terms, beyond which a(52083) = 53724 while for the EKG sequence a(52083) = 26861, this later value being the first prime where the prime-race leader changes; see A007350.

Examples

			a(5) = 8 as a(4) = 6 and 8 is unused and shares a factor with 6. Note that 3 cannot be chosen as 3 is of the form 4*k + 3, and no primes of form 4*k + 1 have yet occurred. This is the first term to differ from A064413.
a(7) = 5 as a(6) = 10 and 5 is unused and shares a factor with 10. This is the first prime of the form 4*k + 1 to occur.
a(9) = 3 as a(8) = 15 and 3 is unused and shares a factor with 15. As a prime of the form 4*k + 1 has occurred, one of the form 4*k + 3 is now allowed.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000.
• Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.
• Wikipedia, Chebyshev's bias.

Crossrefs

Cf. A064413, A007350, A038698, A027748.