A297026 Positions of records in A297025.

0, 1, 2, 3, 5, 7, 11, 22, 23, 46, 47, 94, 167, 283, 359, 718, 719, 1438, 1439, 2878, 2879, 5758, 11516, 23032, 34549, 69098, 138059, 138197, 276394, 552788, 1105576, 2211152, 3316619, 6633238, 11792393, 23584786, 23584787, 47169574, 53065907, 106131814, 212263628, 424527256

Offset: 1

Author

Peter Kagey, Dec 24 2017

Keywords

• nonn

Comments

From David A. Corneth, Dec 24 2017: (Start)

If n > 2 then a(n) <= 2 * a(n - 1). Proof: 2 * a(n - 1) is even. After one iteration of A220096, we get a(n - 1), which gives a record.

If n > 3 and a(n) < 2 * a(n - 1) then a(n) is odd. Proof: if a(n) is even then a(n) / 2 < a(n - 1) is in the sequence. We have k = A297025(a(n - 1)) and k + 1 = A297025(2 * a(n - 1)) hence a(n) can't be the position of a record as a(n - 1) < a(n) < 2 * a(n-1).

If n > 2 and a(n) < 2 * a(n - 1) then a(n) is prime. Proof: This is true for n = 3. For n > 3, a(n) is odd. If a(n) is composite then it has a smallest odd prime factor p >= 3. We have A297025(a(n) / p) < A297025(a(n - 1)) < A297025(a(n)) which is impossible hence in this case, a(n) is prime. (End)

Crossrefs

Cf. A220096, A297025.

Programs

Mathematica

With[{s = Array[Length@ NestWhileList[If[#1 == 1, 0, If[Total[#2[[All, -1]] ] == 1, #1 - 1, #1/#2[[1, 1]] ]] & @@ {#, FactorInteger@ #} &, #, # > 0 &] - 1 &, 2^18, 0] }, FirstPosition[s, #][[1]] - 1 & /@ Union@ FoldList[Max, s]] (* Michael De Vlieger, Dec 24 2017, after Robert G. Wilson v at A297025 *)

PARI

f(n) = if (n==1, 0, isprime(n), n-1, my(d=divisors(n)); d[#d-1]);
nb(n) = my(nb = 0); while (n, n = f(n); nb++); nb;
lista(nn) = {my(rec = - 1); for (n=0, nn, if ((m=nb(n)) > rec, rec = m; print1(n, ", ")););} \\ Michel Marcus, Dec 24 2017

PARI

first(n) = {n = max(n, 2); my(res = vector(n), i = 3, c = 2, m = 1); res[1] = 0; res[2] = 1; while(i <= n, forprime(p = res[i-1] + 1, 2*res[i-1], c = A297025(p); if(c > m, m = c; res[i] = p; i++; next(2))); if(res[i] == 0, res[i] = 2 * res[i-1]; i++; m++)); res}
A220096(n) = if(n == 1, return(0)); my(f = factor(n)); if(vecsum(f[,2])==1, n-1, n / f[1,1])
A297025(n) = my(t); while(n, t++; n = A220096(n)); t \\ David A. Corneth, Dec 24 2017

Extensions

a(29)-a(33) from Michel Marcus, Dec 24 2017

More terms from David A. Corneth, Dec 24 2017