A306582 a(n) is the least integer k such that the remainder of k modulo p is strictly increasing over the first n primes.

0, 2, 4, 34, 52, 194, 502, 1138, 4042, 5794, 5794, 62488, 798298, 5314448, 41592688, 483815692, 483815692, 5037219688, 18517814158, 18517814158, 19566774820732, 55249201504132, 1257253598786974, 6743244322196288, 24165921989926702, 24165921989926702, 5346711077171356252, 47449991406350138602, 278545375679341352084, 5604477496256287791854

Offset: 1

Author

Charlie Neder, Jun 03 2019

Keywords

• nonn
• hard

Comments

If "strictly increasing" is replaced with "nondecreasing", this sequence becomes A000004.

Trivially, a(n) <= A002110(n)-2. Equality only holds for n = 0.

Examples

			  a(n) modulo 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
  ==== ==================================================
     0        0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0, ...
     2        0, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2, ...
     4        0, 1, 4, 4,  4,  4,  4,  4,  4,  4,  4, ...
    34        0, 1, 4, 6,  1,  8,  0, 15, 11,  5,  3, ...
    52        0, 1, 2, 3,  8,  0,  1, 14,  6, 23, 21, ...
   194        0, 2, 4, 5,  7, 12,  7,  4, 10, 20,  8, ...
   502        0, 1, 2, 5,  7,  8,  9,  8, 19,  9,  6, ...
  1138        0, 1, 3, 4,  5,  7, 16, 17, 11,  7, 22, ...
  4042        0, 1, 2, 3,  5, 12, 13, 14, 17, 11, 12, ...
  5794        0, 1, 4, 5,  8,  9, 14, 18, 21, 23, 28, ...
		

Crossrefs

Cf. A000004, A002110, A306612, A325057.

Programs

PARI

isok(k, n) = {my(last = -1, cur); for (i=1, n, cur = k % prime(i); if (cur <= last, return (0)); last = cur;); return (1);}
a(n) = {my(k=0); while(!isok(k, n), k++); k;} \\ Michel Marcus, Jun 04 2019

Python

from sympy import prime
def A306582(n):
    plist, rlist, x = [prime(i) for i in range(1,n+1)], [0]*n, 0
    while True:
        for i in range(n-1):
            if rlist[i] >= rlist[i+1]:
                break
        else:
            return x
        for i in range(n):
            rlist[i] = (rlist[i] + 1) % plist[i]
        x += 1 # Chai Wah Wu, Jun 15 2019

Extensions

a(16)-a(20) from Daniel Suteu, Jun 03 2019

a(21)-a(23) from Giovanni Resta, Jun 16 2019

a(24)-a(27) from Bert Dobbelaere, Jun 22 2019

a(28)-a(30) from Bert Dobbelaere, Sep 05 2019