A387080 a(1)=1, a(2)=3; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

1, 3, 6, 4, 2, 8, 10, 5, 15, 12, 9, 18, 16, 14, 7, 21, 24, 22, 20, 25, 30, 28, 26, 13, 39, 36, 34, 32, 38, 19, 57, 54, 52, 50, 48, 46, 44, 42, 40, 35, 45, 33, 27, 51, 17, 68, 66, 64, 62, 60, 58, 56, 49, 63, 69, 23, 92, 90, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70

Offset

1,2

Comments

This is a variant of A386482 that begins with 1,3 instead of 1,2.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Michael De Vlieger, Log log scatterplot of a(n) in red and A386482(n) in blue, n = 1..2^20.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where the latter represents powerful numbers that are not prime powers.

Mathematica

Block[{c, j, k, m, p, r, nn},
  nn = 2^12; c[_] := False; m[_] := 1; j = 2; c[1] = c[2] = True; r = 1;
  {1}~Join~Monitor[Most@ Reap[Do[
    If[PrimePowerQ[j],
      Set[{p, k, m}, {#1, #1^(#2 - 1), #1^(#2 - 1)}] & @@
        FactorInteger[j][[1]]; While[And[c[k*p], k != 0], k--];
        If[k == 0, k = m; While[c[k*p], k++]]; k *= p,
      k = j - 1; While[And[Or[c[k], CoprimeQ[j, k]], k != 1], k--];
        If[k == 1, k += j; While[Or[c[k], CoprimeQ[j, k] ], k++] ] ];
    If[Mod[j, 2] == Mod[k, 2], r++, Sow[r]; r = 1];
    Set[{c[k], j}, {True, k}], {n, 3, nn}] ][[-1, 1]], n] ]

Prog

(Python)
from math import gcd
from itertools import count, islice
def A387080_gen(): # generator of terms
    m, mset, c = 3, {1, 3}, 2
    yield from (1, 3)
    while True:
        for i in range(m-1, c-1, -1):
            if i not in mset and gcd(i, m)>1:
                yield i
                break
        else:
            for i in count(max(c, m+1)):
                if i not in mset and gcd(i, m)>1:
                    yield i
                    break
        m = i
        mset.add(i)
        while c in mset:
            mset.remove(c)
            c += 1
A387080_list = list(islice(A387080_gen(), 30)) # Chai Wah Wu, Sep 10 2025

Crossrefs

Cf. A386482.

Sequence in context: A359572 A011307 A243625 * A321872 A221363 A245943

Adjacent sequences: A387077 A387078 A387079 * A387081 A387082 A387083

Keyword

nonn

Author

Geoffrey Caveney and Michael De Vlieger, Aug 16 2025

Status

approved