A337136 -id:A337136 - cp's OEIS frontend

A358444 a(1) = 1, a(2) = 2; for n > 2, a(n) = smallest positive number which has not appeared that has a common factor with a(n-2)^2 + a(n-1)^2.

1, 2, 5, 29, 4, 857, 10, 734549, 539562233501, 6, 12433, 15, 8, 17, 353, 12, 124753, 13, 14, 20, 16, 18, 22, 24, 25, 1201, 26, 41, 2357, 28, 5556233, 37, 30, 2269, 39, 32, 35, 52, 3929, 40, 15438641, 82, 45, 65, 34, 5381, 78, 50, 36, 38, 42, 44, 46, 48, 51, 3, 9, 21, 27, 33, 54, 55, 91

Offset: 1

Scott R. Shannon, Nov 16 2022

nonn

The majority of terms are concentrated along or just above the line a(n) = n, resulting in 51 fixed points in the first 5000 terms. However, some terms are much larger because the sum of the squares of the previous two terms is a prime number.

Conjecture: the sequence is a permutation of the positive integers.

			a(5) = 4 as a(3)^2 + a(4)^2 = 25 + 841 = 866, and 4 is the smallest unused number that shares a factor with 866.
a(9) = 539562233501 as a(7)^2 + a(8)^2 = 100 + 539562233401 = 539562233501, which is a prime number.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 10000 terms where a(n) is within 10% of n. The green line is a(n) = n.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..24857. a(24858) is a multiple of a prime factor of 2345424289569907866042152579118178340801^2 + 24922^2.
Michael De Vlieger, Table of n, a(n) for n = 1..65536.

Cf. A337136, A347594, A098550, A336957.

nn = 120; c[] = False; q[] = 1; Do[Set[{a[i], c[i], q[i]}, {i, True, 2}], {i, 2}]; i = a[1]^2; j = a[2]^2; Do[k = i + j; s = FactorInteger[k][[All, 1]]; Do[(m = q[#]; While[c[# m], m++]; q[#] = m; If[# m < k, k = # m]) &[s[[n]]], {n, Length[s]}]; Set[{a[n], c[k], i, j}, {k, True, j, k^2}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Nov 17 2022 *)

A360931 a(1) = 2, a(2) = 3; for n > 2, a(n) is the smallest number greater than 1 that has not appeared such that |a(n) - a(n-1)| has a common factor with a(n-2).

Original entry on oeis.org

2, 3, 5, 8, 13, 7, 20, 6, 4, 10, 12, 14, 11, 9, 31, 16, 47, 15, 62, 17, 19, 36, 55, 21, 26, 23, 25, 48, 18, 22, 24, 28, 30, 32, 27, 29, 35, 64, 34, 38, 40, 42, 37, 33, 70, 43, 39, 82, 46, 44, 50, 52, 54, 41, 45, 86, 51, 49, 58, 65, 53, 63, 116, 56, 60, 66, 57, 59, 68, 127, 61, 188, 249, 67, 73, 140

Offset: 1

Scott R. Shannon, Feb 25 2023

nonn

In the first 100000 terms the fixed points are 10, 16, 42, 52, 66; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers > 1.

			a(6) = 7 as |7 - a(5)| = |7 - 13| = 6 which shares a common factor with a(4) = 8.

Scott R. Shannon, Image for n = 1..100000. The green line is a(n) = n.

Cf. A098550, A336957, A337136, A359557, A353239.

nn = 120; c[] = False; Array[Set[{a[#], c[# + 1]}, {# + 1, True}] &, 2]; i = 2; j = 3; u = 4; Do[k = u; While[Or[c[k], CoprimeQ[i, #]] &[Abs[k - j]], 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, Feb 26 2023 *)

A363198 a(n) = n for n <= 3; for n >= 4, a(n) is the smallest positive integer that has not appeared previously in this sequence and shares a factor with a(n-1) + a(n-2) + a(n-3).

Original entry on oeis.org

1, 2, 3, 4, 6, 13, 23, 7, 43, 73, 9, 5, 12, 8, 10, 14, 16, 15, 18, 21, 20, 59, 22, 101, 24, 27, 19, 25, 71, 30, 26, 127, 33, 28, 32, 31, 35, 34, 36, 39, 109, 38, 40, 11, 89, 42, 44, 45, 131, 46, 37, 48, 262, 347, 51, 50, 49, 52, 151, 54, 257, 55, 56, 58, 65

Offset: 1

Yifan Xie, May 21 2023

Conjecture: This sequence is a permutation of the natural numbers.

Yifan Xie, Table of n, a(n) for n = 1..2000

Cf. A064413, A337136.

a[1] = 1; a[2] = 2; a[3] = 3; a[n_] := a[n] = Module[{s, i = 1}, s = a[n - 1] + a[n - 2] + a[n - 3]; While[MemberQ[a /@ Range[1, n - 1], i] || GCD[s, i] == 1, i++]; i];
Table[a[n], {n, 1, 65}] (* Robert P. P. McKone, Dec 30 2023 *)

lista(nn) = {my(v = [1, 2, 3]); for(n=4, nn, my(t=1); while(prod(X=1, n-1, v[X]-t)==0 || gcd(v[n-3]+v[n-2]+v[n-1], t)==1, t++); v=concat(v, t)); v;}

from math import gcd
a = [1, 2, 3]
t = set(a)
def next_element():
    s = a[-1] + a[-2] + a[-3]
    n = 1
    while n in t or gcd(s, n) == 1:
        n += 1
    return n
def a_seq(ul):
    for _ in range(4, ul + 1):
        nn = next_element()
        a.append(nn)
        t.add(nn)
    return a
print(a_seq(65)) # Robert P. P. McKone, Dec 30 2023

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A358444 a(1) = 1, a(2) = 2; for n > 2, a(n) = smallest positive number which has not appeared that has a common factor with a(n-2)^2 + a(n-1)^2.

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A360931 a(1) = 2, a(2) = 3; for n > 2, a(n) is the smallest number greater than 1 that has not appeared such that |a(n) - a(n-1)| has a common factor with a(n-2).

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A363198 a(n) = n for n <= 3; for n >= 4, a(n) is the smallest positive integer that has not appeared previously in this sequence and shares a factor with a(n-1) + a(n-2) + a(n-3).

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