A343404 For any number n with representation (d_w, ..., d_1) in primorial base, a(n) is the least number m such that m mod prime(k) = d_k for k = 1..w (where prime(k) denotes the k-th prime number).

0, 1, 4, 1, 2, 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, 120, 15, 190, 85, 50, 155, 36, 141, 106, 1, 176, 71, 162, 57, 22, 127, 92, 197, 78, 183, 148, 43, 8, 113, 204, 99, 64, 169, 134, 29, 30, 135, 100, 205

Offset: 0

Author

Rémy Sigrist, Apr 14 2021

Keywords

• nonn
• base

Comments

Leading zeros in primorial base expansions are ignored.

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

Examples

			For n = 42 :
- the expansion of 42 in primary base is "1200",
- so a(42) mod 2 = 0 => a(42) = 2*t for some t >= 0,
     a(42) mod 3 = 0 => a(42) = 6*u for some u >= 0,
     a(42) mod 5 = 2 => a(42) = 12 + 30*v for some v >= 0,
     a(42) mod 7 = 1 => a(42) = 162 + 210*w for some w >= 0,
- we choose w=0 so as to minimize the value,
- hence a(42) = 162.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Wikipedia, Chinese remainder theorem
• Index entries for sequences related to primorial base

Crossrefs

Cf. A002110, A079276, A143293, A235168, A343405 (fixed points).

Programs

PARI

a(n) = { my (v=Mod(0,1)); forprime (p=2, oo, if (n==0, return (lift(v)), v=chinese(v, Mod(n, p)); n\=p)) }

Formula

a(n) = 1 iff n belongs to A143293.

a(n) = n iff n belongs to A343405.

a(n) < A002110(k) for any n < A002110(k) and k >= 0.

a(A002110(k)) = A079276(k+1) * A002110(k) for any k >= 0.