A379785 -id:A379785 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 11 2025

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).

			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, ...

Cf. A379899, A091237.

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}.

Original entry on oeis.org

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

Offset: 1

Robert C. Lyons, Jan 12 2025

			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, ...

Cf. A091237, A379783, A379784.

See also A379652, A379785.

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 *)

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)

cp's OEIS Frontend

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}.

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