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

1,1

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.

Links

Table of n, a(n) for n=1..30.

Example

   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, ...

Prog

(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

Crossrefs

Cf. A306582.

Sequence in context: A037013 A050069 A219019 * A117587 A359747 A244930

Adjacent sequences: A306609 A306610 A306611 * A306613 A306614 A306615

Keyword

nonn,hard

Author

Charlie Neder, Jun 03 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

Status

approved