A372697 Index k such that A280866(k) = A019565(n) or 0 if A019565(n) does not appear in A280866.

1, 2, 4, 5, 7, 8, 17, 26, 11, 12, 20, 37, 36, 67, 68, 205, 14, 15, 46, 63, 74, 90, 127, 302, 73, 145, 146, 373, 307, 736, 1101, 2126, 23, 22, 47, 76, 75, 121, 122, 364, 78, 176, 177, 510, 343, 842, 1229, 2607, 180, 275, 276, 826, 553, 1387, 1388, 4088, 827, 1878

Offset: 0

Author

Michael De Vlieger, Jul 29 2024

Keywords

• nonn

Comments

Offset matches A019565.

Conjecture: there are no zeros in this sequence, which is equivalent to the conjecture that A280866 is a permutation of natural numbers.

Examples

			Let s = A019565 and let t = A280866.
a(0) = 1 since s(0) = 1 = t(1).
a(1) = 2 since s(1) = 2 = t(2).
a(2) = 4 since s(2) = 3 = t(4).
a(3) = 5 since s(3) = 5 = t(5).
Table relating this sequence to s and t. The last column shows Y if s(n) is divisible by the prime in the heading, otherwise ".":
   n   s(n)  a(n)   2357
  ----------------------
   0     1     1    .
   1     2     2    Y
   2     3     4    .Y
   3     6     5    YY
   4     5     7    ..Y
   5    10     8    Y.Y
   6    15    17    .YY
   7    30    26    YYY
   8     7    11    ...Y
   9    14    12    Y..Y
  10    21    20    .Y.Y
  11    42    37    YY.Y
  12    35    36    .YYY
  13    70    67    Y.YY
  14   105    68    .YYY
  15   210   205    YYYY
  ...
		

Links

• Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color code associated with log(a(n))/log(2) for a(n) <= 4194304. Terms that are either 0 or greater than 4194304 appear blank.

Crossrefs

Cf. A005117, A019565, A280866, A372514.

Programs

Mathematica

nn = 2^14; c[] := False; m[] := 1;
i = 1; j = m[1] = m[2] = 2; c[1] = c[2] = True;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
s = Association[
  Monitor[Reap[
     Do[While[c[Set[k, #   m[#]]], m[#]++] &[f[i * j]/f[i]];
      If[SquareFreeQ[k],
        Sow[Total[2^(-1 + PrimePi[FactorInteger[k][[All, 1]]])] -> n] ];
      Set[{c[k], i, j}, {True, j, k}], {n, 3, nn}] ][[-1, 1]], n]];
TakeWhile[{1, 2}~Join~Array[If[KeyExistsQ[s, #], Lookup[s, #], 0] &, Floor@ Sqrt[nn], 2], # > 0 &]