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

%I A372514 #39 Apr 12 2025 11:22:45
%S A372514 1,2,4,5,7,8,17,26,11,12,20,37,36,72,73,207,14,15,43,68,42,106,107,
%T A372514 310,47,151,152,442,294,745,746,2227,23,22,44,53,52,130,114,386,83,
%U A372514 188,156,519,189,884,754,2573,115,269,270,816,387,1405,1406,4134,563,1954
%N A372514 Index k such that A280864(k) = A019565(n) or 0 if A019565(n) does not appear in A280864.
%C A372514 Offset matches A019565.
%C A372514 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.
%C A372514 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.
%H A372514 Michael De Vlieger, <a href="/A372514/b372514.txt">Table of n, a(n) for n = 0..478</a>
%H A372514 Michael De Vlieger, <a href="/A372514/a372514.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) <= 262144. Terms that are either 0 or greater than 262144 appear blank.
%F A372514 a(2^k) > 0 and a(2*m+1) > 0, consequences of Theorem 1 in A280864.
%e A372514 Let s = A019565 and let t = A280864.
%e A372514 a(0) = 1 since s(0) = 1 = t(1).
%e A372514 a(1) = 2 since s(1) = 2 = t(2).
%e A372514 a(2) = 4 since s(2) = 3 = t(4).
%e A372514 a(3) = 5 since s(3) = 5 = t(5).
%e A372514 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 A372514    n   s(n)  a(n)   2357
%e A372514   ----------------------
%e A372514    0     1     1    .
%e A372514    1     2     2    Y
%e A372514    2     3     4    .Y
%e A372514    3     6     5    YY
%e A372514    4     5     7    ..Y
%e A372514    5    10     8    Y.Y
%e A372514    6    15    17    .YY
%e A372514    7    30    26    YYY
%e A372514    8     7    11    ...Y
%e A372514    9    14    12    Y..Y
%e A372514   10    21    20    .Y.Y
%e A372514   11    42    37    YY.Y
%e A372514   12    35    36    .YYY
%e A372514   13    70    72    Y.YY
%e A372514   14   105    73    .YYY
%e A372514   15   210   207    YYYY
%e A372514   ...
%t A372514 nn = 2^13; r = s = 1; c[_] := False;
%t A372514 rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
%t A372514 a = Monitor[Reap[Do[w = GCD[r, s]; k = m = r/w;
%t A372514     While[Or[c[k], ! CoprimeQ[w, k] ], k += m]; Sow[k]; c[k] = True;
%t A372514     s = r; r = rad[k], {i, nn}]][[-1, 1]], i];
%t A372514 Array[FirstPosition[a, Times @@ Prime@ Position[Reverse[IntegerDigits[#, 2]], 1][[All, 1]] ][[1]] &, 61, 0]
%Y A372514 Cf. A005117, A019565, A280864, A372697.
%K A372514 nonn
%O A372514 0,2
%A A372514 _Michael De Vlieger_, Jul 29 2024