A372697 - OEIS

%I #32 Jul 30 2024 15:25:12

%S 1,2,4,5,7,8,17,26,11,12,20,37,36,67,68,205,14,15,46,63,74,90,127,302,

%T 73,145,146,373,307,736,1101,2126,23,22,47,76,75,121,122,364,78,176,

%U 177,510,343,842,1229,2607,180,275,276,826,553,1387,1388,4088,827,1878

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

%C Offset matches A019565.

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

%H Michael De Vlieger, <a href="/A372697/a372697.png">Fan style binary tree showing a(n)</a>, 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.

%e Let s = A019565 and let t = A280866.

%e a(0) = 1 since s(0) = 1 = t(1).

%e a(1) = 2 since s(1) = 2 = t(2).

%e a(2) = 4 since s(2) = 3 = t(4).

%e a(3) = 5 since s(3) = 5 = t(5).

%e 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 ".":

%e    n   s(n)  a(n)   2357

%e   ----------------------

%e    0     1     1    .

%e    1     2     2    Y

%e    2     3     4    .Y

%e    3     6     5    YY

%e    4     5     7    ..Y

%e    5    10     8    Y.Y

%e    6    15    17    .YY

%e    7    30    26    YYY

%e    8     7    11    ...Y

%e    9    14    12    Y..Y

%e   10    21    20    .Y.Y

%e   11    42    37    YY.Y

%e   12    35    36    .YYY

%e   13    70    67    Y.YY

%e   14   105    68    .YYY

%e   15   210   205    YYYY

%e   ...

%t nn = 2^14; c[_] := False; m[_] := 1;

%t i = 1; j = m[1] = m[2] = 2; c[1] = c[2] = True;

%t f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];

%t s = Association[

%t   Monitor[Reap[

%t      Do[While[c[Set[k, #   m[#]]], m[#]++] &[f[i * j]/f[i]];

%t       If[SquareFreeQ[k],

%t         Sow[Total[2^(-1 + PrimePi[FactorInteger[k][[All, 1]]])] -> n] ];

%t       Set[{c[k], i, j}, {True, j, k}], {n, 3, nn}] ][[-1, 1]], n]];

%t TakeWhile[{1, 2}~Join~Array[If[KeyExistsQ[s, #], Lookup[s, #], 0] &, Floor@ Sqrt[nn], 2], # > 0 &]

%Y Cf. A005117, A019565, A280866, A372514.

%K nonn

%O 0,2

%A _Michael De Vlieger_, Jul 29 2024