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

0,2

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.

Links

Table of n, a(n) for n=0..57.

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.

Example

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
  ...

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 &]

Crossrefs

Cf. A005117, A019565, A280866, A372514.

Sequence in context: A141493 A103118 A346140 * A372514 A262969 A158029

Adjacent sequences: A372694 A372695 A372696 * A372698 A372699 A372700

Keyword

nonn

Author

Michael De Vlieger, Jul 29 2024

Status

approved