A362883 a(n) = A055498(n) - A055500(n).

-1, 0, 0, 0, 0, 4, 6, 12, 24, 42, 68, 122, 208, 336, 552, 904, 1464, 2378, 3848, 6232, 10090, 16338, 26446, 42802, 69252, 112072, 181332, 293412, 474762, 768190, 1242960, 2011162, 3254150, 5265324, 8519478, 13784866, 22304378, 36089262, 58393658, 94482964, 152876664

Offset: 0

Author

Philip Baciaz, May 07 2023

Keywords

• sign

Comments

After the initial zeros, {a(n)} seems approximately linear on a log scale (not a surprise with the prime number theorem in mind?), a(n-1)/a(n) seems to converge to the golden ratio (A001622), and 1/a(5) + 1/a(6) + ... + 1/a(n) seems to converge to 0.60086367622...

Crossrefs

Cf. A055500, A055498, A001622.

Programs

Maple

b:= proc(n, t) option remember; `if`(n<3, n+1, (h-> `if`(t=1,
      prevprime(h), nextprime(h)))(t+b(n-1, t)+b(n-2, t)))
    end:
a:= n-> b(n,-1)-b(n,1):
seq(a(n), n=1..50);  # Alois P. Heinz, May 12 2023

Mathematica

b[n_, t_] := b[n, t] = If[n < 3, n+1, Function[h, If[t == 1, NextPrime[h, -1], NextPrime[h]]][t + b[n-1, t] + b[n-2, t]]];
a[n_] := If[n == 0, -1, b[n-1, -1] - b[n-1, 1]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 19 2024, after Alois P. Heinz *)