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

1,1

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).

Links

Table of n, a(n) for n=1..17.

Example

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, ...

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 *)

Crossrefs

Cf. A379899, A091237.

See also A379652, A379785.

Sequence in context: A292775 A282024 A356617 * A086519 A090689 A145476

Adjacent sequences: A379780 A379781 A379782 * A379784 A379785 A379786

Keyword

nonn,more

Author

N. J. A. Sloane, Jan 11 2025

Extensions

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

Status

approved