A360407 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).

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, 3, 28, 13, 8, 23, 24, 9, 4, 19, 14, 29, 0, 105, 70, 175, 140, 35, 126, 21, 196, 91, 56, 161, 42, 147, 112, 7, 182, 77, 168, 63, 28, 133, 98, 203, 84, 189, 154, 49

Offset: 0

Author

Rémy Sigrist, Feb 06 2023

Keywords

• nonn
• base
• tabf

Comments

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

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

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

Examples

			Table T(n, k) begins:
    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, 3, 28, 13, 8, 23, 24, 9, 4, 19, 14, 29;
    ...
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..2558
• Wikipedia, Chinese remainder theorem
• Index entries for sequences related to primorial base

Crossrefs

See A343404 for a similar sequence.

Cf. A002110, A057588, A070826.

Programs

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) }

Formula

T(n, 0) = 0.

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

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

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