A356432 a(1) = 1; for n > 1, a(n) is the smallest positive number not occurring earlier that shares a factor with a(n-1) plus the number of divisors of a(n-1).

1, 2, 4, 7, 3, 5, 14, 6, 8, 9, 10, 12, 15, 19, 18, 16, 21, 20, 13, 24, 22, 26, 25, 28, 17, 38, 27, 31, 11, 39, 43, 30, 32, 34, 36, 33, 37, 42, 35, 45, 48, 29, 62, 40, 44, 46, 50, 49, 52, 54, 56, 58, 60, 51, 55, 59, 61, 57, 122, 63, 23, 65, 66, 64, 71, 73, 69, 146, 68, 70, 72, 74, 75, 78, 76, 41

Offset

1,2

Comments

The sequence is conjectured to be a permutation of the positive integers. In the first 250000 terms there are twenty-three fixed points: 1, 2, 12, 16, 27 ..., 2279, 5401, 7339. It is possibly no more exist although this is unknown.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.

Example

a(7) = 14 as a(6) = 5 which has two divisors, and 14 is the smallest unused number that shares a factor with 5 + 2 = 7.

Maple

A[1]:= 1; S:= {$2..5000}:
for i from 2 do
  found:= false;
  t:= A[i-1] + numtheory:-tau(A[i-1]);
  for s in S do
    if igcd(s, t) > 1 then
      A[i]:= s;
      found:= true;
      S:= S minus {s};
      break
    fi
  od;
  if not found then break fi;
od:
seq(A[j], j=1..i-1); # Robert Israel, Jan 16 2023

Crossrefs

Cf. A356430, A356431, A000005, A354960, A348086.

Sequence in context: A373786 A359005 A256998 * A303641 A137282 A139696

Adjacent sequences: A356429 A356430 A356431 * A356433 A356434 A356435

Keyword

nonn

Author

Scott R. Shannon, Aug 07 2022

Status

approved