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

Author

Michael De Vlieger, Jul 29 2024

Keywords

• nonn

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.

Examples

			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
  ...
		

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.

Crossrefs

Cf. A005117, A019565, A280864, A372697.

Programs

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]

Formula

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