A339670 -id:A339670 - cp's OEIS frontend

A339671 a(1) = 1, a(2) = 2; for n>2, a(n) = smallest number not already used that shares a prime factor with a(n-1) and has a prime factor not in a(n-2).

1, 2, 4, 6, 3, 15, 5, 10, 8, 12, 9, 21, 7, 14, 16, 18, 20, 22, 11, 33, 24, 26, 13, 39, 27, 30, 25, 35, 28, 32, 34, 17, 51, 36, 38, 19, 57, 42, 40, 44, 46, 23, 69, 45, 48, 50, 52, 54, 56, 49, 63, 60, 55, 65, 70, 58, 29, 87, 66, 62, 31, 93, 72, 64, 68, 74, 37, 111, 75, 78, 76, 80, 82, 41, 123, 81

Offset: 1

Scott R. Shannon, Dec 12 2020

nonn

Inspired by A064413 and A336957. The terms show a similar pattern to A064413, and like that sequence they are likely a permutation of the positive integers. Many terms also match the values in A169837. For example a(17)=20 to a(115)=111 (shifted by an index of 1) are the same, but then differ again before more matches occurr.

See A339670 for a similar sequence where the prime factor rules are reversed.

			a(4) = 6 as a(3) = 4 = 2*2 and a(2) = 2, thus a(4) must contain 2 as a prime factor but must also contain a prime factor other than 2. The lowest unused number matching these criteria is 2*3 = 6.
a(6) = 15 as a(5) = 3 and a(4) = 6 = 2*3, thus a(6) must contain 3 as a prime factor but must also contain a prime factor other than 2 and 3. The lowest unused number matching these criteria is 3*5 = 15. This is the first term that differs from A064413.

Cf. A339670, A064413, A169837, A336957, A098550, A255582, A020639.

Block[{a = {1, 2}, b = {}, c = {2}, p, k}, Do[k = 2; While[Nand[FreeQ[a, k], IntersectingQ[c, Set[p, FactorInteger[k][[All, 1]]]], Length@ Complement[p, Intersection[b, p]] > 0], k++]; AppendTo[a, k]; b = c; c = p, 75]; a] (* Michael De Vlieger, Dec 12 2020 *)

A339192 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - n if a(n) is nonnegative, not already in the sequence, and gcd(a(n-1),n) > 1 or gcd(a(n-2),n) > 1. Otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 64, 40, 15, 41, 14, 42, 71, 101, 132, 100, 67, 33, 68, 32, 69, 31, 70, 30, 71, 29, 72, 28, 73, 27, 74, 26, 75, 125, 176, 124, 177, 123, 178, 122, 179, 121, 180, 120, 181, 119, 56, 120, 55, 121, 188, 256, 325, 255

Offset: 0

Scott R. Shannon, Dec 07 2020

nonn

This sequence is a variation of the Recamán sequence A005132 where the same rules apply except an additional restriction is added whereby a(n) = a(n-1) - n can occur only if gcd(a(n-1),n) > 1 or gcd(a(n-2),n) > 1, where gcd is the greatest common divisor. This additional restriction is inspired by the selection rules of A336957 and A098550.
Initially the sequence terms show a similar pattern to the Recamán sequence. However after about 1.5 million terms they begin to predominantly oscillate between two or a small number of values and the pattern of arching lines is no longer present. See the linked images.
It is unclear if all values are eventually visited; numerous small values like 4 and 5 have not occurred after 50 million terms.

			a(4) = 2. As gcd(a(3),4) = gcd(6,4) = 2 > 1, and as 6 - 4 = 2 has not occurred previously, a(4) = 2.
a(23) = 64. a(22) = 41, and 41 - 23 = 18 has not occurred previously. However as gcd(41,23) = 1 and gcd(a(21),23) = gcd(63,23) = 1, both additional criteria for subtraction fail, thus a(23) = a(22) + 23 = 41 + 23 = 64. This is the first term that differs from the standard Recamán sequence A005132.
a(57) = 179. a(56) = 122, and 122 - 57 = 65 has not occurred previously. However as gcd(122,57) = 1 and gcd(a(55),57) = gcd(178,57) = 1, both additional criteria for subtraction fail, thus a(57) = a(56) + 57 = 122 + 57 = 179. This is the first term where n is a composite, less than the last term, and a(n-1) - n is available, but due to the gcd requirements the next term is forced to be a(n-1) + n.

Scott R. Shannon, Image of the terms for n=0 to 10000. The values form a pattern very similar to the Recamán sequence.
Scott R. Shannon, Image of the terms for n=0 to 2000000. Notice the change in behavior after about 1.5 million terms.
Scott R. Shannon, Image of the terms for n=0 to 10000000.
Scott R. Shannon, Image of the terms for n=0 to 50000000.

Cf. A339557, A339670, A339671, A005132, A336957, A098550.

cp's OEIS Frontend

A339671 a(1) = 1, a(2) = 2; for n>2, a(n) = smallest number not already used that shares a prime factor with a(n-1) and has a prime factor not in a(n-2).

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