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A306582
a(n) is the least integer k such that the remainder of k modulo p is strictly increasing over the first n primes.
5
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.
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
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