A355702 -id:A355702 - cp's OEIS frontend

A356017 The fixed points of A355702.

1, 2, 3, 12, 30, 54, 60, 92, 165, 186, 190, 247, 249, 287, 446, 671, 674, 910, 1089, 6310, 6335, 6815, 6844, 9116, 9243, 10215, 11805, 12234, 12244, 12249, 12467, 12522, 13074, 13481, 13556, 13564, 14751, 15505, 15597, 16158, 16220, 16232, 16242, 16306, 17036, 18069, 18578, 21742

Offset: 1

Scott R. Shannon, Jul 23 2022

nonn

In the first 500000 terms of A355702 there are seventy-three fixed points. It is plausible that the total number of fixed points is finite although this is unknown. See A355702 for further details.

Cf. A355702.

A363162 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 divisors as a(n-2) + a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 12, 14, 15, 9, 18, 11, 13, 20, 21, 16, 17, 22, 24, 26, 28, 33, 19, 34, 23, 35, 36, 25, 27, 38, 39, 40, 29, 44, 31, 45, 46, 48, 50, 51, 32, 37, 52, 41, 54, 55, 43, 56, 57, 47, 58, 30, 62, 63, 49, 65, 42, 53, 68, 59, 61, 60, 64, 69, 72, 74, 75, 67, 76, 77, 80, 71, 73, 82

Offset: 1

Scott R. Shannon, Jul 06 2023

nonn

The terms with different numbers of distinct prime divisors are concentrated along different lines in the graph; see the attached colored image. There are numerous fixed points in the first one million terms, although the last nonprime fixed point is a(n) = 3495. Beyond that there are thirty-one more fixed points all with prime values; it is likely more exist although this is unknown. The sequence is conjectured to be a permutation of the positive integers.

			a(4) = 4 as a(2) + a(3) = 2 + 3 = 5 which has one distinct prime divisor, and 4 is the smallest unused number that has one distinct prime divisor.
a(10) = 12 as a(8) + a(9) = 8 + 10 = 18 which has two distinct prime divisors, and 12 is the smallest unused number that has two distinct prime divisors.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Colored image of the first one million terms. The terms with 1,2,3,4,5,6 or 7 distinct prime divisors are colored across the spectrum from red to violet respectively. The white line is a(n) = n.

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

nn = 120; c[] := False; f[x] := PrimeNu[x]; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; i = 1; j = s = 2; u = 3; Do[k = u; s = f[i + j]; While[Or[c[k], f[k] != s], k++]; Set[{a[n], c[k], i, j}, {k, True, j, k}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jul 08 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

A356017 The fixed points of A355702.

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A363162 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 divisors as a(n-2) + a(n-1).

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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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