A360407 - OEIS

%I #9 Feb 08 2023 14:47:05

%S 0,0,1,0,3,4,1,2,5,0,15,10,25,20,5,6,21,16,1,26,11,12,27,22,7,2,17,18,

%T 3,28,13,8,23,24,9,4,19,14,29,0,105,70,175,140,35,126,21,196,91,56,

%U 161,42,147,112,7,182,77,168,63,28,133,98,203,84,189,154,49

%N Irregular table T(n, k), n >= 0, k = 0..A002110(n)-1, read by rows; for any k with primorial base expansion (d_n, ..., d_1), T(n, k) is the least number t such that t mod prime(u) = d_u for u = 1..n (where prime(u) denotes the u-th prime number).

%C When computing T(n, k), we pad the primorial base expansion of k with leading zeros so as to have n digits.

%C The Chinese remainder theorem ensures that this sequence is well defined and provides a way to compute it.

%C The n-th row is a permutation of 0..A002110(n)-1.

%H Rémy Sigrist, <a href="/A360407/b360407.txt">Table of n, a(n) for n = 0..2558</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Chinese_remainder_theorem">Chinese remainder theorem</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F T(n, 0) = 0.

%F T(n, 1) = A070826(n) for any n > 0.

%F T(n, A002110(n) - 1) = A057588(n) for any n > 0.

%F T(n, k) + T(n, A002110(n) - 1 - k) = A002110(n) - 1.

%e Table T(n, k) begins:

%e     0;

%e     0, 1;

%e     0, 3, 4, 1, 2, 5;

%e     0, 15, 10, 25, 20, 5, 6, 21, 16, 1, 26, 11, 12, 27, 22,

%e                   7, 2, 17, 18, 3, 28, 13, 8, 23, 24, 9, 4, 19, 14, 29;

%e     ...

%o (PARI) T(n,k) = { my (t=Mod(0,1)); if (n, forprime (p=2, prime(n), t=chinese(t, Mod(k, p)); k\=p)); lift(t) }

%Y See A343404 for a similar sequence.

%Y Cf. A002110, A057588, A070826.

%K nonn,base,tabf

%O 0,5

%A _Rémy Sigrist_, Feb 06 2023