A306612 a(n) is the least integer k > 2 such that the remainder of -k modulo p is strictly increasing over the first n primes.

3, 4, 7, 8, 16, 16, 157, 157, 16957, 19231, 80942, 82372, 82372, 9624266, 19607227, 118867612, 4968215191, 31090893772, 118903377091, 187341482252, 1784664085208, 12330789708022, 68016245854132, 68016245854132, 10065964847743822, 74887595879692807, 1825207861455319267, 98403562254816509476, 283462437415903129597, 2126598918934702375802

Offset: 1

Author

Charlie Neder, Jun 03 2019

Keywords

• nonn
• hard

Comments

0, 1, and 2 satisfy this condition for all p, so this sequence starts at 3. The growth of this sequence is much more irregular than that of the companion sequence A306582.

Examples

			   a(n) modulo 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
  ===== ==================================================
      3        1, 0, 2, 4,  8, 10, 14, 16, 20, 26, 28, ...
      4        0, 2, 1, 3,  7,  9, 13, 15, 19, 25, 27, ...
      7        1, 2, 3, 0,  4,  6, 10, 12, 16, 22, 24, ...
      8        0, 1, 2, 6,  3,  5,  9, 11, 15, 21, 23, ...
     16        0, 2, 4, 5,  6, 10,  1,  3,  7, 13, 15, ...
    157        1, 2, 3, 4,  8, 12, 13, 14,  4, 17, 29, ...
  16957        1, 2, 3, 4,  5,  8,  9, 10, 17,  8,  0, ...
  19231        1, 2, 4, 5,  8,  9, 13, 16, 20, 25, 20, ...
  80942        0, 1, 3, 6,  7,  9, 12, 17, 18, 26, 30, ...
		

Crossrefs

Cf. A306582.

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=3); while(!isok(k, n), k++); k;} \\ Michel Marcus, Jun 04 2019

Python

from sympy import prime
def A306612(n):
    plist, x = [prime(i) for i in range(1,n+1)], 3
    rlist = [-x % p for p in plist]
    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(19) from Daniel Suteu, Jun 04 2019

a(20)-a(25) from Giovanni Resta, Jun 16 2019

a(26)-a(27) from Bert Dobbelaere, Jun 22 2019

a(28)-a(30) from Bert Dobbelaere, Sep 04 2019