A380130 For n >= 2, let b(n) = 1 if A379784(n) is 3 mod 4, 0 if A379784(n) is 1 mod 4; form the RUNS transform of {b(n), n >= 2}.

1, 6, 13, 34, 87, 229, 581, 1591, 4268, 11637, 31944, 88526, 246105, 688982, 1936129, 5463517, 15470445

Offset: 1

Author

Robert C. Lyons, Jan 12 2025

Keywords

• nonn
• more

Examples

			A379784 begins 1, 5, 3, 7, 11, 19, 23, 31, 13, 17, 29, 37, ..., and the {b(n), n >= 2} sequence begins 0, 1, 1, 1, 1, 1, 1, 0, ..., whose RUNS transform is 1, 6, ...
		

Crossrefs

Cf. A091237, A379783, A379784.

See also A379652, A379785.

Programs

Mathematica

nn = 2^19; c[_] := True; q = 0; j = r = 1; s = 4;
Monitor[Rest@ Reap[Do[m = j + s;
  While[Set[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 13 2025 *)

Python

from sympy import primefactors
prev_a379784 = 1
prev_b = -1
b_run = 0
a379784_set = set([prev_a379784])
seq = []
max_seq_len = 17
while len(seq) < max_seq_len:
    c = prev_a379784
    done = False
    while not done:
        c = c + 4
        factors = primefactors(c)
        for factor in factors:
            if factor not in a379784_set:
                a379784_set.add(factor)
                if factor % 4 == 3:
                    b = 1
                else:
                    b = 0
                if prev_b >= 0:
                    if b == prev_b:
                        b_run += 1
                    else:
                        seq.append(b_run)
                        b_run = 1
                else:
                    b_run = 1
                prev_b = b
                prev_a379784 = factor
                done = True
                break
print(seq)