A343404 -id:A343404 - cp's OEIS frontend

A343405 Numbers k such that A343404(k) = k.

0, 1, 5, 6, 11, 12, 17, 18, 23, 24, 29, 36, 65, 72, 101, 108, 137, 144, 173, 209, 210, 419, 420, 629, 630, 839, 840, 1049, 1050, 1259, 1260, 1469, 1470, 1679, 1680, 1889, 1890, 2099, 2100, 2309, 2939, 4200, 5670, 7140, 8609, 10079, 11340, 12810, 14280, 15749

Offset: 1

Rémy Sigrist, Apr 14 2021

If m belongs to this sequence, then the number obtained by removing the leading digit from the expansion of m in primary base also belongs to this sequence.
So the terms of the sequence can be mapped on a tree with root 0 (see illustration in Links section).
Is this sequence infinite?

			A343404(2099) = 2099, so 2099 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Illustration of the terms < 2*3*5*7*11
Rémy Sigrist, PARI program for A343405
Index entries for sequences related to primorial base

Cf. A266181, A319599, A343404.

PARI
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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).

Original entry on oeis.org

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

Rémy Sigrist, Feb 06 2023

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.

			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;
    ...

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

See A343404 for a similar sequence.

Cf. A002110, A057588, A070826.

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) }

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.

cp's OEIS Frontend

A343405 Numbers k such that A343404(k) = k.

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