A372514 Index k such that A280864(k) = A019565(n) or 0 if A019565(n) does not appear in A280864.

1, 2, 4, 5, 7, 8, 17, 26, 11, 12, 20, 37, 36, 72, 73, 207, 14, 15, 43, 68, 42, 106, 107, 310, 47, 151, 152, 442, 294, 745, 746, 2227, 23, 22, 44, 53, 52, 130, 114, 386, 83, 188, 156, 519, 189, 884, 754, 2573, 115, 269, 270, 816, 387, 1405, 1406, 4134, 563, 1954

Offset

0,2

Comments

Offset matches A019565.

Based on Selcoe's comment in A280864 regarding k in sequences S_r = { k = m*r : rad(m) | r }, squarefree r > 1, appearing in order. The appearance of r itself introduces the lineage S_r, followed by lpf(r)*r, etc., if A280864 is a permutation of natural numbers.

Conjecture: there are no zeros in this sequence, which is equivalent to the conjecture that A280864 is a permutation of natural numbers. Minor corollary: a(127) > 2^18.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..478

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) <= 262144. Terms that are either 0 or greater than 262144 appear blank.

Formula

a(2^k) > 0 and a(2*m+1) > 0, consequences of Theorem 1 in A280864.

Example

Let s = A019565 and let t = A280864.
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    72    Y.YY
  14   105    73    .YYY
  15   210   207    YYYY
  ...

Mathematica

nn = 2^13; r = s = 1; c[_] := False;
rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
a = Monitor[Reap[Do[w = GCD[r, s]; k = m = r/w;
    While[Or[c[k], ! CoprimeQ[w, k] ], k += m]; Sow[k]; c[k] = True;
    s = r; r = rad[k], {i, nn}]][[-1, 1]], i];
Array[FirstPosition[a, Times @@ Prime@ Position[Reverse[IntegerDigits[#, 2]], 1][[All, 1]] ][[1]] &, 61, 0]

Crossrefs

Cf. A005117, A019565, A280864, A372697.

Sequence in context: A103118 A346140 A372697 * A262969 A158029 A217378

Adjacent sequences: A372511 A372512 A372513 * A372515 A372516 A372517

Keyword

nonn

Author

Michael De Vlieger, Jul 29 2024

Status

approved