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A297026
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Positions of records in A297025.
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1
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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
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OFFSET
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1,3
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COMMENTS
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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)
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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