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%I #54 Sep 05 2019 12:05:59
%S 0,2,4,34,52,194,502,1138,4042,5794,5794,62488,798298,5314448,
%T 41592688,483815692,483815692,5037219688,18517814158,18517814158,
%U 19566774820732,55249201504132,1257253598786974,6743244322196288,24165921989926702,24165921989926702,5346711077171356252,47449991406350138602,278545375679341352084,5604477496256287791854
%N a(n) is the least integer k such that the remainder of k modulo p is strictly increasing over the first n primes.
%C If "strictly increasing" is replaced with "nondecreasing", this sequence becomes A000004.
%C Trivially, a(n) <= A002110(n)-2. Equality only holds for n = 0.
%e a(n) modulo 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
%e ==== ==================================================
%e 0 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 2 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 4 0, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
%e 34 0, 1, 4, 6, 1, 8, 0, 15, 11, 5, 3, ...
%e 52 0, 1, 2, 3, 8, 0, 1, 14, 6, 23, 21, ...
%e 194 0, 2, 4, 5, 7, 12, 7, 4, 10, 20, 8, ...
%e 502 0, 1, 2, 5, 7, 8, 9, 8, 19, 9, 6, ...
%e 1138 0, 1, 3, 4, 5, 7, 16, 17, 11, 7, 22, ...
%e 4042 0, 1, 2, 3, 5, 12, 13, 14, 17, 11, 12, ...
%e 5794 0, 1, 4, 5, 8, 9, 14, 18, 21, 23, 28, ...
%o (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);}
%o a(n) = {my(k=0); while(!isok(k, n), k++); k;} \\ _Michel Marcus_, Jun 04 2019
%o (Python)
%o from sympy import prime
%o def A306582(n):
%o plist, rlist, x = [prime(i) for i in range(1,n+1)], [0]*n, 0
%o while True:
%o for i in range(n-1):
%o if rlist[i] >= rlist[i+1]:
%o break
%o else:
%o return x
%o for i in range(n):
%o rlist[i] = (rlist[i] + 1) % plist[i]
%o x += 1 # _Chai Wah Wu_, Jun 15 2019
%Y Cf. A000004, A002110, A306612, A325057.
%K nonn,hard
%O 1,2
%A _Charlie Neder_, Jun 03 2019
%E a(16)-a(20) from _Daniel Suteu_, Jun 03 2019
%E a(21)-a(23) from _Giovanni Resta_, Jun 16 2019
%E a(24)-a(27) from _Bert Dobbelaere_, Jun 22 2019
%E a(28)-a(30) from _Bert Dobbelaere_, Sep 05 2019
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