A350359 -id:A350359 - cp's OEIS frontend

A352197 Where n appears in A350359, or -1 if n does not appear.

1, 2, 3, 4, 9, 21, 10, 6, 5, 13, 18, 23, 28, 8, 7, 15, 37, 25, 40, 17, 12, 27, 48, 29, 11, 19, 14, 31, 59, 43, 62, 33, 16, 35, 22, 45, 77, 47, 36, 41, 84, 65, 100, 49, 34, 55, 105, 67, 24, 51, 52, 53, 116, 69, 32, 63, 38, 57, 135, 87, 142, 71, 54, 73, 30, 89, 158, 39, 50, 81, 164, 91, 171, 75, 56, 83, 20, 93, 178, 85, 58, 95, 185, 97, 66, 109

Offset: 1

N. J. A. Sloane, Mar 27 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..20000 (first 4326 terms from N. J. A. Sloane)
Rémy Sigrist, PARI program

Cf. A350359, A352198.

PARI
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See Links section.
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A352198 Where n-th prime appears in A350359, or -1 if n-th prime does not appear.

Original entry on oeis.org

2, 3, 9, 10, 18, 28, 37, 40, 48, 59, 62, 77, 84, 100, 105, 116, 135, 142, 158, 164, 171, 178, 185, 210, 221, 234, 240, 245, 262, 269, 287, 308, 318, 321, 334, 342, 365, 371, 384, 405, 412, 415, 430, 451, 459, 462, 478, 494, 501, 502, 537, 547, 550, 580, 587, 606, 621, 622

Offset: 1

N. J. A. Sloane, Mar 27 2022

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 590 terms from N. J. A. Sloane)
Rémy Sigrist, PARI program

Cf. A350359, A352197.

PARI
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See Links section.
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A352950 Lexicographically earliest infinite sequence of distinct nonnegative integers commencing 1,3,5,7 such that any four consecutive terms are pairwise coprime.

Original entry on oeis.org

1, 3, 5, 7, 2, 9, 11, 13, 4, 15, 17, 19, 8, 21, 23, 25, 16, 27, 29, 31, 10, 33, 37, 41, 14, 39, 43, 47, 20, 49, 51, 53, 22, 35, 57, 59, 26, 55, 61, 63, 32, 65, 67, 69, 28, 71, 73, 45, 34, 77, 79, 75, 38, 83, 89, 81, 40, 91, 97, 87, 44, 85, 101, 93, 46, 95, 103, 99, 52, 107, 109, 105

Offset: 1

David James Sycamore, Apr 10 2022

nonn

The pairwise coprime relations found in the first four odd numbers 1,3,5,7 are preserved throughout in any run of four consecutive terms.
a(4n+5) is always even (and < a(4n+2)); n>=0.
The plot exhibits two distinct rays at first (upper/odd, lower/even), with no terms divisible by 6 until a(229), at which point the even ray switches to producing just 28 multiples of 6 until a(337)=168. At this point the original even ray is re-established, the odd ray divides into two (quasi-parallel) rays, and no further multiples of 6 are seen. Therefore it seems very unlikely that the sequence is a permutation of the nonnegative integers.
Primes p other than p = 2 appear in their natural order.

			3,5,7 are pairwise coprime and 2 is the smallest unused number coprime to all of them, therefore a(5)=2.

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, highlighting primes in green, numbers divisible by 6 in gold, other even numbers in red, odd numbers divisible by 3 in blue.

Cf. A085229, A103683, A350359.

ina := proc(n) false end: # adapted from code for A103683
a := proc (n) option remember; local k;
if n < 5 then k := 2*n-1
else for k from 2 while ina(k) or igcd(k, a(n-1)) <> 1 or igcd(k, a(n-2)) <> 1 or igcd(k, a(n-3)) <> 1
do
end do
end if; ina(k):= true; k
end proc;
seq(a(n), n = 1 .. 100);

Block[{a, c, k, m, u, nn}, nn = 86; c[] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 3, 5, 7}]; u = 2; Do[k = u; m = LCM @@ Array[a[i - #] &, 3]; 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, 5, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Apr 12 2022 *)

from math import gcd
from itertools import islice
def agen(): # generator of terms
    aset, b, c, d = {1, 3, 5, 7}, 3, 5, 7
    yield from [1, b, c, d]
    while True:
        k = 1
        while k in aset or any(gcd(t, k) != 1 for t in [b, c, d]): k+= 1
        b, c, d = c, d, k
        aset.add(k)
        yield k
print(list(islice(agen(), 107))) # Michael S. Branicky, Apr 10 2022

A352199 a(0)=0, a(1)=1, a(2)=2; thereafter, a(n) is smallest number m not yet in the sequence such that the binary expansions of m and a(n-2) have a 1 in common, but the 1's in m are disjoint from the 1's in a(n-1) and a(n-3).

Original entry on oeis.org

0, 1, 2, 5, 10, 4, 8, 20, 9, 6, 33, 18, 32, 14, 96, 3, 48, 7, 16, 11, 80, 12, 64, 13, 66, 17, 34, 21, 40, 65, 42, 68, 24, 69, 26, 36, 130, 37, 74, 49, 72, 52, 136, 19, 128, 22, 160, 15, 192, 23, 224, 25, 288, 27, 100, 129, 260, 131, 28, 35, 76, 161, 84, 162, 88

Offset: 0

N. J. A. Sloane, Mar 26 2022

A set-theory analog of A350359. This has the same relationship to A350359 as A115510 does to the EKG sequence A064413, as A252867 does to the Yellowstone permutation A098550, and as A338833 does to the Enots Wolley sequence A336957.

An equivalent definition in terms of sets: S(0) = {}, S(1) = {1}, S(2) = {1,2}; thereafter S(n) is the smallest set (different from the S(i) already defined) of positive integers such that S(n) meets S(n-2) but is disjoint from S(n-1) and S(n-3).

			After a(4) = 10 = 1010_2, a(5) = 4 = 100_2, a(6) = 8 = 1000_2, a(7) must have the form ...?010?_2, and the smallest missing number of that form is 20 = 10100_2 = 20.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A064413, A098550, A115510, A252867, A338833, A350359.

{ s=0; for (n=1, #a=vector(65), if (n<=3, a[n]=n-1, for (v=0, oo, if (!bittest(s,v) && bitand(v,a[n-2]) && !bitand(v,bitor(a[n-3],a[n-1])), a[n]=v; break))); s+=2^a[n]; print1(a[n]", ")) } \\ Rémy Sigrist, Mar 27 2022

cp's OEIS Frontend

A352197 Where n appears in A350359, or -1 if n does not appear.

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A352198 Where n-th prime appears in A350359, or -1 if n-th prime does not appear.

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PARI

A352950 Lexicographically earliest infinite sequence of distinct nonnegative integers commencing 1,3,5,7 such that any four consecutive terms are pairwise coprime.

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A352199 a(0)=0, a(1)=1, a(2)=2; thereafter, a(n) is smallest number m not yet in the sequence such that the binary expansions of m and a(n-2) have a 1 in common, but the 1's in m are disjoint from the 1's in a(n-1) and a(n-3).

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PARI