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A387078 Run lengths of A386482(n) mod 2 == n mod 2.

%I A387078 #7 Aug 16 2025 01:19:13
%S A387078 1,3,2,4,2,3,3,5,3,4,2,8,5,3,4,4,2,11,4,2,2,22,16,5,3,1,2,12,6,31,14,
%T A387078 4,3,8,3,28,2,37,14,10,12,9,2,41,7,61,24,24,2,134,71,51,97,3,2,127,69,
%U A387078 39,15,64,55,56,26,100,37,32,40,33,2,440,107,196,391
%N A387078 Run lengths of A386482(n) mod 2 == n mod 2.
%C A387078 Let S = A386482.
%C A387078 Beginning with S(481) = 948, there are 100 consecutive even terms in S. Starting with S(730076) = 1026330, there are 100869 consecutive even terms in S.
%H A387078 Michael De Vlieger, <a href="/A387078/b387078.txt">Table of n, a(n) for n = 1..183</a> (run lengths available given 2^20 terms of S).
%e A387078 S begins as follows, grouping odd terms in brackets [], and even in parentheses ():
%e A387078    [1], (2, 4, 6), [3, 9], (12, 10, 8, 14), [7, 21], (18, 16, 20), [15, 5, 25], ...
%e A387078 This sequence takes run lengths in the order they appear, therefore a(1) = 1, a(2) = 3, a(3) = 2, a(4) = 4, a(5) = 2, etc. Hence a(n) for odd n pertains to run lengths of odd terms in S, while a(n) for even n pertains to run lengths of even terms in same.
%t A387078 Block[{c, j, k, m, p, r, nn},
%t A387078   nn = 2^12; c[_] := False; m[_] := 1; j = 2; c[1] = c[2] = True; r = 1;
%t A387078   {1}~Join~Monitor[Most@ Reap[Do[
%t A387078     If[PrimePowerQ[j],
%t A387078       Set[{p, k, m}, {#1, #1^(#2 - 1), #1^(#2 - 1)}] & @@
%t A387078         FactorInteger[j][[1]]; While[And[c[k*p], k != 0], k--];
%t A387078         If[k == 0, k = m; While[c[k*p], k++]]; k *= p,
%t A387078       k = j - 1; While[And[Or[c[k], CoprimeQ[j, k]], k != 1], k--];
%t A387078         If[k == 1, k += j; While[Or[c[k], CoprimeQ[j, k] ], k++] ] ];
%t A387078     If[Mod[j, 2] == Mod[k, 2], r++, Sow[r]; r = 1];
%t A387078     Set[{c[k], j}, {True, k}], {n, 3, nn}] ][[-1, 1]], n] ]
%Y A387078 Cf. A386482.
%K A387078 nonn
%O A387078 1,2
%A A387078 _Michael De Vlieger_, Aug 15 2025