A353006 -id:A353006 - cp's OEIS frontend

A353075 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) * a(n-2) + |a(n-1) - a(n-2)|.

1, 2, 3, 7, 5, 37, 14, 541, 8101, 223, 23, 73, 13, 1009, 11, 12097, 46, 22, 4, 6, 8, 10, 12, 16, 18, 15, 9, 21, 24, 26, 20, 28, 30, 32, 34, 25, 859, 35, 17, 613, 69, 42841, 39, 1713601, 19, 92, 27, 2549, 38, 43, 33, 1429, 115, 44, 42, 36, 40, 48, 50, 52, 54, 45, 51, 57, 60, 49, 65, 55, 63

Offset: 1

Scott R. Shannon, Apr 22 2022

nonn

The sequence produces numerous groupings of primes. For example a(3) to a(16) contains thirteen primes in fourteen terms, a(80) to a(102) contains fourteen primes in twenty-three terms. The sequences is conjectured to be a permutation of the positive integers.

			a(5) = 5 as a(4)*a(3)+|a(4)-a(3)| = 7*3+|7-3| = 25, and 5 is the smallest unused number that shares a factor with 25.

Michael De Vlieger, Table of n, a(n) for n = 1..5000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..5000, showing primes in red.

Cf. A353006, A353082, A352867, A352774, A352768.

nn = 69; c[] = 0; a[1] = c[1] = 1; a[2] = c[2] = 2; u = 3; Do[m = #1 #2 + Abs[#2 - #1] & @@ {a[i - 2], a[i - 1]}; If[PrimeQ[m], k = 1; While[c[k m] != 0, k++]; k = k m, k = u; While[Nand[c[k] == 0, ! CoprimeQ[m, k]], k++]]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, May 02 2022 *)

A353082 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a divisor of (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 13, 8, 7, 15, 11, 26, 37, 21, 16, 185, 39, 14, 25, 33, 29, 31, 6, 925, 19, 12, 217, 41, 22, 57, 35, 23, 24, 47, 71, 48, 17, 65, 32, 97, 43, 10, 53, 63, 20, 83, 103, 30, 73, 4429, 18, 401, 383, 28, 137, 55, 64, 51, 115, 128, 27, 101, 74, 45, 119, 82, 67, 149, 36, 113, 77

Offset: 1

Scott R. Shannon, Apr 22 2022

The sequence is finite, the final term being a(160)=143. See the examples. The maximum value is a(110)=35941 while the fixed points are 1, 2, 3, 8, 11, 67.

			a(6) = 9 as (a(5)+a(4))*|a(5)-a(4)| = (4+5)*|4-5| = 9, and 9 is the smallest unused number that is a divisor of 9. This is the first term that differs from A353006.
a(159) = 144, a(160) = 143. Therefore a(161) would be the smallest unused number that divides (143+144)*|143-144| = 287. The divisors are 1, 7, 41, 287, all of which have already appeared, so the sequence terminates.

Cf. A353006, A337136, A352867, A352790, A350927.

cp's OEIS Frontend

A353075 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) * a(n-2) + |a(n-1) - a(n-2)|.

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