A352187 -id:A352187 - cp's OEIS frontend

A352200 a(0)=0, a(1)=1; thereafter, a(n) is the smallest number m not yet in the sequence such that the binary expansions of m and a(n-1) have a 1 in the same position, but the positions of the 1's in the binary expansions of m and a(n-2) are disjoint, except that the second condition is ignored if it would imply that no choice for m were possible.

0, 1, 3, 2, 6, 4, 5, 9, 8, 10, 7, 17, 16, 18, 11, 12, 20, 19, 33, 32, 34, 14, 13, 49, 48, 21, 15, 40, 96, 64, 65, 23, 22, 24, 41, 35, 66, 68, 28, 25, 67, 38, 36, 29, 26, 98, 37, 129, 128, 130, 27, 44, 100, 80, 144, 131, 39, 52, 88, 72, 30, 50, 97, 69, 132, 136, 42, 51, 81, 76, 46, 146, 145, 45, 70, 82, 56, 137, 71, 54, 152, 73, 99, 134, 140, 57

Offset: 0

N. J. A. Sloane, Mar 26 2022

The second condition is ignored precisely when the positions of the 1's in a(n-1) are a subset of the 1's in a(n-2).

This is a set-theory analog of A352187.

			a(0)=0 and a(1)=1=1_2 are given.
a(2) = 3 = 11_2 is disjoint from a(0) and intersects a(1).
a(3) = 2 = 10_2 is disjoint from a(1) and intersects a(2).
Now there is no choice for a(4) that meets both conditions, so we ignore the no-intersection-with-a(n-2) condition, and take a(4) = 6 = 110_2.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Maple program

This completes a set of four pairs of sequences: (A064413 and A115510), (A098550 and A252867), (A336957 and A338833), (A352187 and this sequence, A352200).

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A358026 Let G(n) = gcd(a(n-2),a(n-1)), a(1)=1, a(2)=2, a(3)=3. Thereafter if G(n) = 1, a(n) is the least novel m sharing a divisor with both a(n-2) and a(n-1). If G(n) > 1 and every prime divisor of a(n-1) also divides a(n-2), a(n) is the least m prime to both a(n-1) and a(n-2). Otherwise a(n) is the least novel multiple of any prime divisor of a(n-1) which does not divide a(n-2).

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 10, 8, 7, 14, 12, 9, 11, 33, 15, 20, 16, 13, 26, 18, 21, 28, 22, 44, 17, 34, 24, 27, 19, 57, 30, 25, 23, 115, 35, 42, 32, 29, 58, 36, 39, 52, 38, 76, 31, 62, 40, 45, 48, 46, 69, 51, 68, 50, 55, 66, 54, 37, 74, 56, 49, 41, 287, 63, 60, 64, 43

Offset: 1

David James Sycamore, Oct 25 2022

nonn

Conjectured to be a permutation of the positive integers with the primes in natural order, and primes are the slowest numbers to appear (as in A352187).

			a(4) = 6, the least novel number sharing a factor with both 2 and 3.
a(5) = 4, the least novel multiple of 2, which divides a(4) but does not divide a(3).
Since every prime dividing a(5)=4 also divides a(4)=6, a(6)=5 the least novel term prime to 3 and 6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and other prime powers in gold.

Cf. A064413, A336957, A352187, A357963.

nn = 67; c[] = False; q[] = 1; u = 4; Do[(Set[{a[n], c[n]}, {n, True}]; q[n]++), {n, u - 1}]; Do[m = FactorInteger[a[n - 1]][[All, 1]]; f = Select[m, CoprimeQ[#, a[n - 2]] &]; Which[Length[f] == PrimeNu[a[n - 1]], Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, Flatten@ Outer[Times, m, FactorInteger[a[n - 2]][[All, 1]] ] ], Last], Length[f] == 0, k = u; While[Nand[! c[k], CoprimeQ[a[n - 2], k], CoprimeQ[a[n - 1], k]], k++]; If[k == u, While[c[u], u++]], True, Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MinimalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, f], Last] ]; Set[{a[n], c[k]}, {k, True}], {n, 4, nn}]; Array[a, nn] (* Michael De Vlieger, Oct 25 2022 *)

More terms from Michael De Vlieger, Oct 25 2022

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