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

%I A372697 #32 Jul 30 2024 15:25:12
%S A372697 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 A372697 73,145,146,373,307,736,1101,2126,23,22,47,76,75,121,122,364,78,176,
%U A372697 177,510,343,842,1229,2607,180,275,276,826,553,1387,1388,4088,827,1878
%N A372697 Index k such that A280866(k) = A019565(n) or 0 if A019565(n) does not appear in A280866.
%C A372697 Offset matches A019565.
%C A372697 Conjecture: there are no zeros in this sequence, which is equivalent to the conjecture that A280866 is a permutation of natural numbers.
%H A372697 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 A372697 Let s = A019565 and let t = A280866.
%e A372697 a(0) = 1 since s(0) = 1 = t(1).
%e A372697 a(1) = 2 since s(1) = 2 = t(2).
%e A372697 a(2) = 4 since s(2) = 3 = t(4).
%e A372697 a(3) = 5 since s(3) = 5 = t(5).
%e A372697 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 A372697    n   s(n)  a(n)   2357
%e A372697   ----------------------
%e A372697    0     1     1    .
%e A372697    1     2     2    Y
%e A372697    2     3     4    .Y
%e A372697    3     6     5    YY
%e A372697    4     5     7    ..Y
%e A372697    5    10     8    Y.Y
%e A372697    6    15    17    .YY
%e A372697    7    30    26    YYY
%e A372697    8     7    11    ...Y
%e A372697    9    14    12    Y..Y
%e A372697   10    21    20    .Y.Y
%e A372697   11    42    37    YY.Y
%e A372697   12    35    36    .YYY
%e A372697   13    70    67    Y.YY
%e A372697   14   105    68    .YYY
%e A372697   15   210   205    YYYY
%e A372697   ...
%t A372697 nn = 2^14; c[_] := False; m[_] := 1;
%t A372697 i = 1; j = m[1] = m[2] = 2; c[1] = c[2] = True;
%t A372697 f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
%t A372697 s = Association[
%t A372697   Monitor[Reap[
%t A372697      Do[While[c[Set[k, #   m[#]]], m[#]++] &[f[i * j]/f[i]];
%t A372697       If[SquareFreeQ[k],
%t A372697         Sow[Total[2^(-1 + PrimePi[FactorInteger[k][[All, 1]]])] -> n] ];
%t A372697       Set[{c[k], i, j}, {True, j, k}], {n, 3, nn}] ][[-1, 1]], n]];
%t A372697 TakeWhile[{1, 2}~Join~Array[If[KeyExistsQ[s, #], Lookup[s, #], 0] &, Floor@ Sqrt[nn], 2], # > 0 &]
%Y A372697 Cf. A005117, A019565, A280866, A372514.
%K A372697 nonn
%O A372697 0,2
%A A372697 _Michael De Vlieger_, Jul 29 2024