A382091 -id:A382091 - cp's OEIS frontend

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 4k+1 and 4k+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

Scott R. Shannon, Mar 09 2025

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

			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.

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.

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

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 4k+1 and 4k+3 for all terms a(1)..a(n) never differs by more than 1.

Original entry on oeis.org

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

Scott R. Shannon, May 24 2025

nonn

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.

			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.

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.

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

cp's OEIS Frontend

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 4k+1 and 4k+3 for all terms a(1)..a(n) never differs by more than 1.

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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 4k+1 and 4k+3 for all terms a(1)..a(n) never differs by more than 1.

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cp's OEIS Frontend

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.

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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.

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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 4k+1 and 4k+3 for all terms a(1)..a(n) never differs by more than 1.

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 4k+1 and 4k+3 for all terms a(1)..a(n) never differs by more than 1.