A381902 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 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, 4, 6, 8, 10, 5, 15, 3, 12, 16, 20, 14, 26, 13, 39, 9, 30, 25, 35, 7, 28, 32, 34, 17, 51, 18, 40, 22, 50, 24, 38, 52, 44, 55, 60, 58, 29, 87, 21, 70, 64, 68, 46, 74, 37, 111, 33, 75, 45, 65, 78, 36, 80, 48, 82, 41, 123, 42, 91, 104, 56, 100, 62, 31

Offset

1,2

Comments

Unlike the EKG sequence A064413 the prime terms are not in their natural order, and the terms preceding and following such terms can be large multiples of the prime. The terms overall are distributed over multiple lines, with the primes falling on at least two lines; see the attached colored image. Due to the term selection rules numbers which have a sum of prime factor exponents for prime factors of the form 4*k+1 and 4*k+3 which differ by 3 or more can never appear, the smallest such number being 27.

In the first 100000 terms the fixed points are 1, 2, 88, 118, 304, 786, 826. It is likely no more exist.

There are five dominant lines on the graph of the first 100000 terms. They can be characterized as follows, from the highest sloped L1 to the lowest sloped L5, considering terms within 1% of the fitted equations. The approximate slopes of the five lines are 2.1284, 1.476, 1.4190, 1.06845, and 0.70947, so that the normalized slopes of L1, L3, L4 and L5 are 3, 2, 3/2 and 1. L5 has essentially has only prime terms, while the others essentially have none. The 5 lines encompass approx. 97% of terms in the range 50K-100K. - Bill McEachen, Aug 21 2025

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the first 100000 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 thin green line is a(n) = n.

Wikipedia, Chebyshev's bias.

Example

a(5) = 8 as the total number of prime factors of the form 4*k+1 and 4*k+3 for the first four terms is 0 and 1 respectively, thus a(5) cannot contain a single prime factor of the form 4*k+3. This eliminates 3 as a candidate, leaving 8 as the smallest available number that has no such prime factors and shares a factor with a(4) = 6. This is the first term to differ from A064413.
a(7) = 5 as the total number of prime factors of the form 4*k+1 and 4*k+3 for the first six terms is 1 and 1 respectively, thus a term can be chosen that contains a single odd prime factor, and 5 is the smallest unused term that shares a factor with a(6) = 10.

Crossrefs

Cf. A064413, A382091, A007350, A038698, A027748.

Sequence in context: A115950 A077127 A382091 * A082191 A319806 A055952

Adjacent sequences: A381899 A381900 A381901 * A381903 A381904 A381905

Keyword

nonn

Author

Scott R. Shannon, Mar 09 2025

Status

approved