A356850 -id:A356850 - cp's OEIS frontend

A360394 Intersection of A026430 and A360392.

3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 80, 82, 84, 86, 88, 90, 93, 95, 98, 100, 102, 105, 107, 109, 111, 113, 116, 118, 120, 123, 125, 128, 130, 132, 134, 136, 138

Offset: 1

Clark Kimberling, Feb 05 2023

This is the first of four sequences that partition the positive integers. Starting with a general overview, suppose that u = (u(n)) and v = (v(n)) are increasing sequences of positive integers.  Let u' and v' be their complements, and assume that the following four sequences are infinite:
(1)  u ^ v = intersection of u and v (in increasing order);
(2)  u ^ v';
(3)  u' ^ v;
(4)  u' ^ v'.
Every positive integer is in exactly one of the four sequences. The limiting densities of these four sequences are 4/9, 2/9, 2/9, and 1/9, respectively (and likewise for A360402-A360405).
For A360394, u, v, u', v', are sequences obtained from the Thue-Morse sequence, A026430, as follows:
u = A026530 = (1,3,5,6,8,9,10, 12, ... ) = partial sums of A026430
u' = A356133 = (2,4,7,11,13,17, 20, ... ) = complement of u
v = u + 1 = A285954, except its initial 1
v' = complement of v.

			(1)  u ^ v = (3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, ...) =  A360394
(2)  u ^ v' = (1, 6, 9, 15, 19, 24, 27, 31, 36, 42, 45, 51, 55, 60, ...) =  A360395
(3)  u' ^ v = (7, 11, 17, 20, 25, 29, 32, 38, 43, 47, 53, 56, 62, ...) = A360396
(4)  u' ^ v' = (2, 4, 13, 22, 34, 40, 49, 58, 64, 76, 85, 94, 106, ...) = A360397

Cf. A026430, A356133, A360392, A360393, A356850-A360405.

z = 400;
u = Accumulate[1 + ThueMorse /@ Range[0, z]];   (* A026430 *)
u1 = Complement[Range[Max[u]], u];  (* A356133 *)
v = u + 2 ; (* A360392 *)
v1 = Complement[Range[Max[v]], v];  (* A360393 *)
Intersection[u, v]     (* A360394 *)
Intersection[u, v1]    (* A360395 *)
Intersection[u1, v]    (* A360396 *)
Intersection[u1, v1]   (* A360397 *)

A356851 a(1) = 1, a(2) = 2, a(3) = 4; for n > 3, a(n) is the smallest positive number not occurring earlier such that a(n) shares a factor with the previous Omega(a(n)) terms.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 12, 15, 5, 10, 20, 14, 7, 21, 28, 35, 30, 25, 40, 45, 50, 18, 22, 8, 16, 24, 26, 13, 39, 52, 65, 78, 60, 33, 11, 44, 55, 66, 70, 34, 17, 51, 68, 85, 102, 90, 38, 19, 57, 76, 95, 114, 110, 46, 23, 69, 92, 115, 138, 130, 58, 29, 87, 116, 145, 174, 150, 62, 31, 93, 124, 155, 186

Offset: 1

Scott R. Shannon, Aug 31 2022

nonn

For n > 250 the terms are concentrated along seven lines, see the linked images. Unlike the other six lines, numbers along the second lowest line are somewhat spread out, and these terms contain all numbers with Omega(a(n)) > 1. The lowest line contains all the primes, while the upper five lines contain terms with Omega(a(n)) = 2, 3, and 4. The primes up to n=100000 occur in their natural order except for 11 and 13 which are switched. The only fixed point beyond the first two terms is 10, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(8) = 15 as Omega(15) = A001222(15) = 2, and 15 shares a factor with the previous two terms, namely a(6) = 9 and a(7) = 12.

Scott R. Shannon, Image of n=1..100000. The green line is a(n) = n.
Scott R. Shannon, Image of n=1..100000 with color. Terms with Omega(a(n)) = 1, 2, 3, 4, 5, >=6 are shown in red, yellow, green, blue, violet, white respectively.

Cf. A064413, A356850 (coprime to previous Omega(a(n)) terms), A001222, A093714, A336957, A000040

A356903 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous tau(a(n)) terms.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 9, 11, 13, 17, 8, 15, 19, 23, 29, 14, 25, 27, 31, 37, 22, 35, 39, 41, 43, 34, 47, 21, 53, 55, 26, 49, 51, 59, 61, 10, 67, 33, 71, 73, 38, 65, 69, 77, 79, 58, 83, 57, 85, 89, 46, 91, 87, 95, 97, 62, 101, 103, 81, 107, 74, 109, 113, 93, 115, 82, 119, 121, 111, 125, 86, 127, 131, 123

Offset: 1

Scott R. Shannon, Sep 03 2022

nonn

The terms are concentrated along various lines that contain numbers with a lowest prime factor of 2, 3 or 5. These lines appear to have a slight upward curvature. However the uppermost line, which has a gradient of ~1.22, contains numbers with all prime factors. See the linked images.

Numbers with a large number of divisors relative to the numbers close to it appear much later in the sequence. For example a(96) = 6, a(1873) = 12, a(2328) = 18, a(192) = 16. The sequence is conjectured to be a permutation of the positive integers although it may take a very large number of terms for some values to appear, e.g., after 500000 terms numbers such as 24, 30, 36 have not occurred. In the same range the longest run of consecutive odd values is seven, while the only fixed points are the first three terms, although it is possible others exist for very large values of n if the smaller terms continue to increase relative to the uppermost line.

			a(7) = 9 as tau(9) = A000005(9) = 3, and 9 is coprime to the previous three terms, namely a(6) = 4, a(5) = 7 and a(4) = 5.

Scott R. Shannon, Image of the first 500000 terms.
Scott R. Shannon, Image of the first 500000 terms with color. The terms with a lowest prime factor of 2 are shown in red, those with 3 are shown in yellow, those with 5 are shown in green, while all others are shown in white.

Cf. A356850, A000005, A356851, A093714, A336957, A000040.

A358277 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous Omega(a(n-1)) terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 10, 13, 12, 17, 14, 15, 19, 16, 23, 18, 25, 29, 20, 21, 31, 22, 27, 35, 26, 33, 37, 24, 41, 28, 43, 30, 47, 32, 53, 34, 39, 49, 38, 45, 59, 36, 61, 40, 67, 42, 71, 44, 65, 51, 46, 55, 57, 52, 73, 48, 79, 50, 77, 69, 58, 83, 54, 85, 89, 56, 97, 60, 101, 62, 63, 95

Offset: 1

Scott R. Shannon, Nov 08 2022

nonn

Unlike A356850 all the terms are concentrated along three straight lines. In the first 100000 terms there are ten fixed points, 1, 2, 3, ..., 27, 57, and it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(4) = 4 as a(3) = 3 and Omega(3) = A001222(3) = 1, and 4 is coprime to the previous one term, namely a(3) = 3.
a(9) = 11 as a(8) = 8 and Omega(8) = A001222(8) = 3, and 11 is coprime to the previous three terms, namely 8, 7, 6.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^10, with records in red, local minima in blue, highlighting primes in green and other prime powers in gold, showing numbers that are neither squarefree nor prime powers in magenta.
Michael De Vlieger, Scalar scatterplot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, composite squarefree numbers in green, products of composite prime powers in magenta, and other numbers in blue.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^10 using a color function showing Omega(a(n-1)).

Cf. A356850, A356903, A001222, A356851, A093714, A336957, A000040.

nn = 90; c[] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 3; j = 2; Do[k = u; m = Times @@ Array[a[n - #] &, PrimeOmega[j]]; While[Nand[! c[k], CoprimeQ[k, m]], k++]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Nov 20 2022 *)

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A360394 Intersection of A026430 and A360392.

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A356851 a(1) = 1, a(2) = 2, a(3) = 4; for n > 3, a(n) is the smallest positive number not occurring earlier such that a(n) shares a factor with the previous Omega(a(n)) terms.

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A356903 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous tau(a(n)) terms.

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A358277 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous Omega(a(n-1)) terms.

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