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

1,2

Comments

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

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

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, ...
  ==== ==================================================
     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, ...

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=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

Crossrefs

Cf. A000004, A002110, A306612, A325057.

Sequence in context: A178811 A099433 A051225 * A103625 A006989 A236399

Adjacent sequences: A306579 A306580 A306581 * A306583 A306584 A306585

Keyword

nonn,hard

Author

Charlie Neder, Jun 03 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

Status

approved