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A379783 For n >= 2, let b(n) = 1 if A379899(n) is 3 mod 4, 0 if A379899(n) is 1 mod 4; form the RUNS transform of {b(n), n >= 2}.

%I A379783 #19 Jan 11 2025 18:41:21
%S A379783 3,7,19,42,116,292,791,2085,5692,15482,42709,118272,329891,923905,
%T A379783 2600458,7344965,20818129
%N A379783 For n >= 2, let b(n) = 1 if A379899(n) is 3 mod 4, 0 if A379899(n) is 1 mod 4; form the RUNS transform of {b(n), n >= 2}.
%C A379783 If instead of A379899 we begin with the primes >= 2 in their natural order, the {b(n), n >= 2} sequence is 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, ..., with RUNS transform 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 2, ..., (a dramatically different sequence, essentially A091237).
%e A379783 A379899 begins 2, 3, 7, 11, 5, 13, 17, 29, 37, 41, 53, 19, ..., and the {b(n), n >= 2} sequence begins 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, ..., whose RUNS transform is 3, 7, 19, 42, ...
%t A379783 nn = 2^20; c[_] := True; j = 3; q = 0; r = 1; s = 4;
%t A379783 Monitor[Reap[
%t A379783   Do[m = j + s;
%t A379783     While[k = SelectFirst[FactorInteger[m][[All, 1]], c]; !IntegerQ[k],
%t A379783       m += s];
%t A379783     c[k] = False; j = k;
%t A379783     If[# == r, q++, r = #; Sow[q]; q = 1] &[(Mod[k, 4] - 1)/2],
%t A379783 {n, nn}] ][[-1, 1]], n] (* _Michael De Vlieger_, Jan 11 2025 *)
%Y A379783 Cf. A379899, A091237.
%Y A379783 See also A379652, A379785.
%K A379783 nonn,more
%O A379783 1,1
%A A379783 _N. J. A. Sloane_, Jan 11 2025
%E A379783 a(10)-a(17) from _Michael De Vlieger_, Jan 11 2025