A308193 -id:A308193 - cp's OEIS frontend

A308190 Number of steps to reach 5 when iterating x -> A111234(x) starting at x=n.

0, 1, 3, 2, 2, 4, 4, 3, 4, 3, 3, 5, 6, 5, 5, 4, 5, 5, 5, 4, 5, 4, 4, 6, 8, 7, 7, 6, 4, 6, 4, 5, 7, 6, 6, 6, 7, 6, 6, 5, 6, 6, 6, 5, 4, 5, 5, 7, 10, 9, 6, 8, 6, 8, 8, 7, 6, 5, 5, 7, 6, 5, 7, 6, 5, 8, 8, 7, 8, 7, 7, 7, 6, 8, 8, 7, 8, 7, 7, 6, 6, 7, 7, 7, 8, 7, 5, 6, 7, 5, 5, 6, 7, 6, 6, 8, 12

Offset: 5

N. J. A. Sloane, Jun 14 2019

nonn

It is easy to show that every number n >= 5 eventually reaches 5. This was conjectured by Ali Sada. For A111234 sends a composite n > 5 to a smaller number, and sends a prime > 5 to a smaller number in two steps. Furthermore no number >= 5 can reach a number less than 5. So all numbers >= 5 eventually reach 5.

Ali Sada, Email to N. J. A. Sloane, Jun 13 2019.

Chai Wah Wu, Table of n, a(n) for n = 5..10000

Cf. A111234, A308191, A308192, A308193, A308194.

a[n_] := Module[{c = 0, x = n, y}, While[x != 5, y = Min[FactorInteger[x][[All, 1]]]; x = y + Quotient[x, y]; c++]; c];
Table[a[n], {n, 5, 100}] (* Jean-François Alcover, Jun 15 2019, from Python *)

from sympy import factorint
def A308190(n):
    c, x = 0, n
    while x != 5:
        y = min(factorint(x))
        x = y + x//y
        c += 1
    return c # Chai Wah Wu, Jun 14 2019

A308191 a(n) = smallest m such that A308190(m) = n, or -1 if no such m exists.

Original entry on oeis.org

5, 6, 8, 7, 10, 16, 17, 30, 29, 54, 53, 102, 101, 198, 197, 390, 389, 774, 773, 1542, 3080, 3079, 6154, 12304, 24604, 36901, 73798, 147592, 295180, 295517, 591030, 1182056, 1574849, 3149694, 4728211, 6299383, 12598762, 25197520, 25197533, 50395062, 100790120, 100790119, 201580234, 403160464, 806320924, 1232145821, 2464291638

Offset: 0

N. J. A. Sloane, Jun 14 2019

nonn

It seems plausible that m exists for all n >= 0.
From Chai Wah Wu, Jun 14 2019: (Start)
All terms are even or prime. If a(n+1) is even, then 2*a(n)-a(n+1) = 4. a(n+1) <= 2*(a(n)-2) and thus m exists for all n >= 0. The proof in the comments of A308193 is applicable for this sequence as well.
If a(n) is prime, then a(n-1) <= a(n) + 1. For the prime terms 7, 17, 29, 53, 101, 197, 389, 773, 3079, 100790119, a(n-1) = a(n) + 1.
(End)

Chai Wah Wu, Table of n, a(n) for n = 0..54

Cf. A111234, A308190.

a(24)-a(41) from Chai Wah Wu, Jun 14 2019
a(42)-a(44) from Chai Wah Wu, Jun 15 2019
a(45)-a(46) from Chai Wah Wu, Jun 16 2019

A308192 Record values in A308190.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54

Offset: 1

N. J. A. Sloane, Jun 14 2019

Cf. A111234, A308190, A308191, A308193.

a(17)-a(36) from Chai Wah Wu, Jun 14 2019
a(37)-a(38) from Chai Wah Wu, Jun 16 2019
a(39)-a(41) from Chai Wah Wu, Jun 17 2019
a(42)-a(45) from Chai Wah Wu, Jun 24 2019

cp's OEIS Frontend

A308190 Number of steps to reach 5 when iterating x -> A111234(x) starting at x=n.

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A308191 a(n) = smallest m such that A308190(m) = n, or -1 if no such m exists.

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A308192 Record values in A308190.

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