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}.

3, 7, 19, 42, 116, 292, 791, 2085, 5692, 15482, 42709, 118272, 329891, 923905, 2600458, 7344965, 20818129

Offset: 1

Author

N. J. A. Sloane, Jan 11 2025

Keywords

• nonn
• more

Comments

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).

Examples

			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, ...
		

Crossrefs

Cf. A379899, A091237.

See also A379652, A379785.

Programs

Mathematica

nn = 2^20; c[_] := True; j = 3; q = 0; r = 1; s = 4;
Monitor[Reap[
  Do[m = j + s;
    While[k = SelectFirst[FactorInteger[m][[All, 1]], c]; !IntegerQ[k],
      m += s];
    c[k] = False; j = k;
    If[# == r, q++, r = #; Sow[q]; q = 1] &[(Mod[k, 4] - 1)/2],
{n, nn}] ][[-1, 1]], n] (* Michael De Vlieger, Jan 11 2025 *)

Extensions

a(10)-a(17) from Michael De Vlieger, Jan 11 2025