A111234 -id:A111234 - 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

A068319 a(n) = if n <= lpf(n)^2 then lpf(n) else a(lpf(n) + n/lpf(n)), where lpf = least prime factor, A020639.

Original entry on oeis.org

1, 2, 3, 2, 5, 5, 7, 5, 3, 7, 11, 5, 13, 3, 5, 7, 17, 11, 19, 5, 7, 13, 23, 3, 5, 5, 5, 7, 29, 17, 31, 11, 3, 19, 5, 5, 37, 7, 7, 13, 41, 23, 43, 3, 11, 5, 47, 5, 7, 5, 5, 7, 53, 29, 7, 17, 13, 31, 59, 11, 61, 3, 3, 19, 11, 5, 67, 5, 5, 37, 71, 7, 73, 7

Offset: 1

Reinhard Zumkeller, Feb 27 2002, Jul 13 2007

n>1: a(n) is prime and a(n)=n iff n is prime.

a(n) = if n <= A088377(n) then A020639(n) else a(A111234(n)).

			a(12)=a(2*6)=a(8)=a(2*4)=a(6)=a(2*3)=a(5)=a(5*1)=5.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A032742.

a068319 n = if n <= spf ^ 2 then spf else a068319 $ spf + div n spf
            where spf = a020639 n
-- Reinhard Zumkeller, Jun 24 2013

lpf[n_] := FactorInteger[n][[1, 1]]; a[n_] := a[n] = If[n <= lpf[n]^2, lpf[n], a[lpf[n] + n/lpf[n]]]; Table[a[n], {n, 1, 74}](* Jean-François Alcover, Dec 21 2011 *)

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

A308193 Indices of records in A308190.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 14 2019

nonn

For terms a(1) through a(16), with one exception, 2*a(n) - a(n+1) is either 4 or 5. Does this pattern continue, and if so, why?
From Chai Wah Wu, Jun 14 2019: (Start)
The pattern does not continue. a(17) = 24604, a(18) = 36901.
Theorem:
1. All terms are even or prime.
2. If a(n+1) is even, then 2*a(n)-a(n+1) = 4.
3. a(n+1) <= 2*(a(n)-2).
Proof: If a(n+1) = x is even, then A111234(x) = 2+x/2 = y. If we assume that x >= 6, then y < x. Thus A308190(x) = A308190(y)+1, i.e., a(n) <= y. If a(n) < y, then A308190(2*(a(n)-2)) = A308190(a(n)) + 1.
Since a(n) is a record value, this means that the next record value is at most at 2*(a(n)-2), i.e., 2*(a(n)-2) < x = a(n+1), a contradiction.
Thus we have shown that if a(n+1) is even, then 2*a(n) = a(n+1)+4.
If a(n+1) = x is an odd composite with smallest prime factor p > 2, then A308190(x) = A308190(y)+1 where y = p+x/p. On the other hand, A308190(2*(y-2)) = A308190(y)+1. Since 2*(y-2) < x, this contradicts the fact that a(n+1) = x is a record value.
(End)

Chai Wah Wu, Table of n, a(n) for n = 1..45

Cf. A111234, A308190, A308191, A308192.

a(17)-a(36) from Chai Wah Wu, Jun 14 2019

a(37)-a(38) from Chai Wah Wu, Jun 16 2019

A140777 a(n) = 2*prime(n) - 4.

Original entry on oeis.org

0, 2, 6, 10, 18, 22, 30, 34, 42, 54, 58, 70, 78, 82, 90, 102, 114, 118, 130, 138, 142, 154, 162, 174, 190, 198, 202, 210, 214, 222, 250, 258, 270, 274, 294, 298, 310, 322, 330, 342, 354, 358, 378, 382, 390, 394, 418, 442, 450, 454, 462, 474, 478, 498, 510, 522

Offset: 1

Leroy Quet, May 29 2008, May 31 2008

A number n is included if (p + n/p) is prime, where p is the smallest prime that divides n. Since all terms of this sequence are even (or otherwise p + n/p would be even and not a prime), p is always 2. So this sequence is the set of all even numbers n where (2 + n/2) is prime.
The entries are also encountered via the bilinear transform approximation to the natural log (unit circle). Specifically, evaluating 2(x-1)/(x+1) at x = 2, 3, 4, ..., the terms of this sequence are seen ahead of each new prime encountered. Additionally, the position of those same primes will occur at the entry positions. For clarity, the evaluation output is 2, 3, 1, 1, 6, 5, 4, 3, 10, 7, 3, 2, 14, 9, 8, 5, 18, 11, ..., where the entries ahead of each new prime are 2, 6, 10, 18, ... . As an aside, the same mechanism links this sequence to A165355. - Bill McEachen, Jan 08 2015
As a follow-up to previous comment, it appears that the numerators and denominators of 2(x-1)/(x+1) are respectively given by A145979 and A060819, but with different offsets. - Michel Marcus, Jan 14 2015
Subset of the union of A017641 & A017593. - Michel Marcus, Sep 01 2020

			The smallest prime dividing 42 is 2. Since 2 + 42/2 = 23 is prime, 42 is included in this sequence.

Cf. A000040, A020639, A040976, A111234, A140775, A140776.
Cf. A060819, A145979, A165355.
Cf. A017641, A017593.

[2*NthPrime(n)-4: n in [1..80]]; // Vincenzo Librandi, Feb 19 2015

A020639 := proc(n) local dvs,p ; dvs := sort(convert(numtheory[divisors](n),list)) ; for p in dvs do if isprime(p) then RETURN(p) ; fi ; od: error("%d",n) ; end: A111234 := proc(n) local p ; p := A020639(n) ; p+n/p ; end: isA140777 := proc(n) RETURN(isprime(A111234(n))) ; end: for n from 2 to 1200 do if isA140777(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 31 2008
seq(2*ithprime(i)-4, i=1..1000); # Robert Israel, Jan 09 2015

fQ[n_] := Block[{p = First@ First@ Transpose@ FactorInteger@ n}, PrimeQ[p + n/p] == True]; Select[ Range[2, 533], fQ@# &] (* Robert G. Wilson v, May 30 2008 *)
Table[2 Prime[n] - 4, {n, 60}] (* Vincenzo Librandi, Feb 19 2015 *)

vector(100, n, 2*prime(n) - 4) \\ Michel Marcus, Jan 09 2015

a(n) = 2*A040976(n). - Michel Marcus, Jan 09 2015

More terms from Robert G. Wilson v and R. J. Mathar, May 30 2008

A308194 Number of steps to reach 5 when iterating x -> A063655(x) starting at x=n.

Original entry on oeis.org

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

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.

Rémy Sigrist, Table of n, a(n) for n = 5..10000

Cf. A063655, A308190, A308195.

b(n) = { my(c=1); fordiv(n, d, if((d*d)>=n, if((d*d)==n, return(2*d), return(c+d))); c=d); (0); } \\ after A063655
a(n) = for (k=0, oo, if (n==5, return (k), n=b(n))) \\ Rémy Sigrist, Jun 14 2019

from sympy import divisors
def A308194(n):
    c, x = 0, n
    while x != 5:
        d = divisors(x)
        l = len(d)
        x = d[(l-1)//2] + d[l//2]
        c += 1
    return c # Chai Wah Wu, Jun 14 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

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

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A068319 a(n) = if n <= lpf(n)^2 then lpf(n) else a(lpf(n) + n/lpf(n)), where lpf = least prime factor, A020639.

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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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A308193 Indices of records in A308190.

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A140777 a(n) = 2*prime(n) - 4.

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A308194 Number of steps to reach 5 when iterating x -> A063655(x) starting at x=n.

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A308192 Record values in A308190.

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