A254078 a(n) is the number of steps after which n variables with increasing value ranges all have equal values when the values of all variables are decreased by 1 at each step and the value is set to the maximum value again when the resulting value would be 0.

4, 10, 58, 58, 418, 838, 2518, 2518, 27718, 27718, 360358, 360358, 360358, 720718, 12252238, 12252238, 232792558, 232792558, 232792558, 232792558, 5354228878, 5354228878, 26771144398, 26771144398, 80313433198, 80313433198, 2329089562798

Offset: 2

Author

Felix Fröhlich, Jan 25 2015

Keywords

• nonn

Comments

The k-th variable can take k+1 different values.

From Charlie Neder, Oct 01 2018: (Start)

a(n) is the smallest k congruent to m-2 modulo m for 2 <= m <= n+1.

Proof: All variables will be equal for the first time precisely when they all are equal to 2, in which case each variable has changed from its maximum value m to 2. Additionally, this k is lcm(2,3,...,m) - 2, since advancing two more steps will return all variables to their maximum values.

Adding a variable that only takes one value {1} results in A070198 (LCM - 1). (End)

Examples

			In case of two variables, the first can take two values (1 and 2) and the second three values (1, 2 and 3). Performing the operation on the variables generates sequences of values 2, 1, 2, 1, 2, 1, ... for first variable and 3, 2, 1, 3, 2, 1, ... for second variable. After four steps, the value of both variables is 2, so a(2) = 4.
		

Links

• Felix Fröhlich, Visual representation of a(3)

Crossrefs

Cf. A070198 (LCM - 1), A003418 (LCM), A075059 (LCM + 1).

Programs

PARI

a(n) = my(v=vector(n, x, x++), w=v, i=0); while(1, if(vecmax(v)==vecmin(v), return(i)); for(k=1, #v, if(v[k]==1, v[k]=w[k], v[k]--)); i++) \\ Felix Fröhlich, Feb 19 2017

Python

from math import gcd
lcm = 2
for n in range(3,53):
  lcm *= n // gcd(lcm,n)
  print(n-1,lcm-2) # Charlie Neder, Oct 02 2018

Formula

a(n) = A003418(n+1) - 2. - Charlie Neder, Oct 02 2018

Extensions

Value of a(6) corrected and more terms from Felix Fröhlich, Mar 25 2015

Illustration and program replaced with improved versions by Felix Fröhlich, Feb 19 2017

Corrected and extended by Charlie Neder, Oct 01 2018