cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A297026 Positions of records in A297025.

Original entry on oeis.org

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

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Author

Peter Kagey, Dec 24 2017

Keywords

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
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