A365642 a(1) = 1, a(2) = 2; a(n) = smallest k such that rad(k) | (a(n-2)+a(n-1)) and k != a(m), m < n, where rad(n) = A007947(n).

1, 2, 3, 5, 4, 9, 13, 8, 7, 15, 11, 16, 27, 43, 10, 53, 21, 32, 2809, 81, 17, 14, 31, 25, 28, 148877, 45, 19, 64, 83, 49, 6, 55, 61, 29, 12, 41, 7890481, 18, 179, 197, 47, 122, 169, 97, 38, 75, 113, 94, 23, 39, 62, 101, 163, 22, 37, 59, 24, 6889, 223, 56, 93, 149

Offset

1,2

Comments

Let s = rad(n). Let r(s) = { k : rad(k) | s }. For example, r(6) = A003586 = {1, 2, 3, 4, 6, 8, 9, 12, ...}.

Terms k in s*{r(s)} appear in this sequence in order. This is to say, for example, that we have 6, then later 12, 18, 24, 36, etc. in the sequence.

Prime powers p^m tend to enter early.

Conjecture: permutation of natural numbers.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10001

Example

a(3) = 3 since a(1)+a(2) = 3, rad(3) | 3 and a(m) != a(3), m < 3.
a(4) = 5 since a(2)+a(3) = 5, rad(5) | 5 and a(m) != a(4), m < 4.
a(5) = 4 since a(3)+a(4) = 8; since a(2) = 2, we try 4 since rad(4) | 8 and a(m) != a(5), m < 5.
a(6) = 9 since a(4)+a(5) = 9; since a(3) = 3, we try 9 since rad(9) | 9 and 9 is not already in the sequence.
a(19) = 2809 since a(17)+(18) = 21+32 = 53; since a(16) = 53, we try 53^2 = 2809, since rad(2809) | 53 and isn't already in the sequence.
a(26) = 148877 since a(24)+a(25) = 25+28 = 53; since a(16) = 53 and a(19) = 53^2, we try 53^3 = 148877 as rad(148877) | 53 and isn't already in the sequence, etc.

Mathematica

nn = 120; c[_] := False; m[_] := 1;
  r[x_] := r[x] = Times @@ FactorInteger[x][[All, 1]];
  i = a[1] = 1; j = a[2] = 2; c[1] = c[2] = True; m[2]++; u = 3;
  Do[If[PrimePowerQ[#],
       While[c[Set[k, #^m[#]]], m[#]++] &[FactorInteger[#][[1, 1]]],
       k = u; While[Or[c[k], ! Divisible[#, rad[k]]], k++]] &[i + j];
    Set[{a[n], c[k], i, j}, {k, True, j, k}];
    If[k == u, While[c[u], u++]], {n, 3, nn}];
  Array[a, nn]

Crossrefs

Cf. A007947.

Sequence in context: A307024 A349493 A124653 * A353082 A085947 A328444

Adjacent sequences: A365639 A365640 A365641 * A365643 A365644 A365645

Keyword

nonn

Author

Michael De Vlieger, Nov 15 2023

Status

approved