A220096 -id:A220096 - cp's OEIS frontend

A297025 Number of iterations of A220096 required to reach 0 starting from n.

0, 1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 6, 6, 5, 5, 6, 5, 6, 6, 6, 7, 8, 6, 5, 7, 5, 7, 8, 6, 7, 6, 7, 7, 6, 6, 7, 7, 7, 7, 8, 7, 8, 8, 6, 9, 10, 7, 6, 6, 7, 8, 9, 6, 7, 8, 7, 9, 10, 7, 8, 8, 7, 7, 7, 8, 9, 8, 9, 7, 8, 7, 8, 8, 6, 8, 7, 8, 9, 8, 6, 9, 10, 8, 7

Offset: 0

Peter Kagey and Alec Jones, Dec 24 2017

nonn

Records occur at indices 0, 1, 2, 3, 5, 7, 11, 22, 23, 46, 47, 94, ... (see A297026).

			For n = 14, a(14) = 6 because six iterations are required to reach zero:
A220096(14) = 7,
A220096(7)  = 6,
A220096(6)  = 3,
A220096(3)  = 2,
A220096(2)  = 1, and
A220096(1)  = 0.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A220096. Positions of records at A297026.

Cf. A056792, A061282, A260112.

g[n_Integer] := If[n == 1, 0, Block[{fi = FactorInteger@ n}, If[Plus @@ (Last@# & /@ FactorInteger@n) == 1, n -1, n/fi[[1, 1]] ]]]; f[n_] := Length@ NestWhileList[g, n, # > 0 &] -1; Array[f, 86, 0] (* Robert G. Wilson v, Dec 24 2017 *)

f(n) = if (n==1, 0, isprime(n), n-1, my(d=divisors(n)); d[#d-1]);
a(n) = my(nb = 0); while (n, n = f(n); nb++); nb; \\ Michel Marcus, Dec 24 2017

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

Peter Kagey, Dec 24 2017

nonn

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)

Cf. A220096, A297025.

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 *)

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

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

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

More terms from David A. Corneth, Dec 24 2017

cp's OEIS Frontend

A297025 Number of iterations of A220096 required to reach 0 starting from n.

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A091934 Number of dual isomorphisms on [ n,n* ].

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A297026 Positions of records in A297025.

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