A352867 -id:A352867 - cp's OEIS frontend

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)|.

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.

A361593 a(1) = 1, a(2) = 2, a(3) = 3; for n > 3, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-3) + a(n-2) + a(n-1) are factors of a(n).

Original entry on oeis.org

1, 2, 3, 6, 11, 10, 9, 30, 7, 46, 83, 34, 163, 70, 267, 20, 357, 322, 699, 1378, 2399, 2238, 6015, 5326, 13579, 6230, 25135, 106, 31471, 14178, 45755, 15234, 75167, 68078, 8341, 151586, 228005, 193966, 573557, 248882, 1016405, 306474, 1571761, 361830, 2240065, 1043414, 3645309, 3464394

Offset: 1

Scott R. Shannon and Michael De Vlieger, Mar 16 2023

nonn

This is a variation of A359557 where the previous three terms are added instead of two. Unlike A359557 the terms here do no rapidly reach a regime where all terms share one or more prime factors, and it is unknown if this ever occurs.

			a(6) = 10 as a(3) + a(4) + a(5) = 3 + 6 + 11 = 20 = 2*2*5, and the smallest unused number containing 2 and 5 as factors is 10.

Michael De Vlieger, Table of n, a(n) for n = 1..750
Michael De Vlieger, Scatterplot of Log_10(a(n)), n = 1..750, showing records in red.
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..600, showing primes in red, prime powers in gold, and squarefree composites in green.

Cf. A359557, A359256, A027748, A064413, A352867, A007947.

nn = 120; c[] = False; q[] = 1;
f[n_] := Times @@ FactorInteger[n][[All, 1]]; t = 3;
Array[Set[{a[#], c[#]}, {#, True}] &, t]; Set[{i, j, k, x}, {a[t - 2],
   a[t - 1], a[t], f[a[t - 2] + a[t - 1] + a[t]]}];
Do[m = q[x];
  While[c[x m], m++];
  m *= x; While[c[x q[x]], q[x]++];
  Set[{a[n], c[m], i, j, k, x}, {m, True, j, k, m, f[j + k + m]}], {n,
t + 1, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 20 2023 *)

A364164 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has the same number of distinct prime factors as the sum of all previous terms.

Original entry on oeis.org

1, 2, 3, 6, 10, 12, 14, 15, 18, 4, 20, 30, 21, 42, 60, 66, 22, 24, 70, 78, 84, 90, 26, 28, 33, 34, 35, 36, 102, 105, 5, 38, 110, 39, 7, 210, 114, 120, 126, 330, 390, 420, 130, 132, 138, 140, 462, 510, 150, 546, 570, 154, 40, 44, 45, 156, 8, 165, 630, 660, 168, 170, 174, 9, 46, 48, 690

Offset: 1

Scott R. Shannon, Jul 12 2023

nonn

In the first 20000 terms the largest value is a(14889) = 15825810 which contains eight distinct prime factors. In the same range there are 593 terms that are prime, the last being a(19985) = 4339, while the smallest number not to appear is 4349. It is likely all numbers eventually appear.

			a(3) = 3 as the sum of all previous terms is 1 + 2 = 3 which contains one distinct prime factor, and 3 is the smallest unused number that also contains one distinct prime factor.
a(6) = 12 as the sum of all previous terms is 1 + 2 + 3 + 6 + 10 = 22 which contains two distinct prime factors, and 12 is the smallest unused number that also contains two distinct prime factors.

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

Cf. A001221, A363162, A355702, A355647, A355649, A352867.

cp's OEIS Frontend

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)|.

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A361593 a(1) = 1, a(2) = 2, a(3) = 3; for n > 3, a(n) is the smallest positive number which has not appeared such that all the distinct prime factors of a(n-3) + a(n-2) + a(n-1) are factors of a(n).

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A364164 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that has the same number of distinct prime factors as the sum of all previous terms.

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