A079276 -id:A079276 - cp's OEIS frontend

A341805 Numbers k such that (product of first k primes)-1 is divisible by the (k+1)-th prime.

0, 2, 4, 9823712

Offset: 1

Jeppe Stig Nielsen, Feb 20 2021

			4 is a member because 2*3*5*7-1 (product of first 4 primes, minus one) is divisible by the 5th prime, 11.
9823712 is a member because 2*3*5*...*176078267-1 is divisible by 176078293, where 176078267 is the 9823712th prime.

Cf. A079276, A081618, A338543, A341804, A341812.

isok(k) = ((vecprod(primes(k)) - 1) % prime(k+1)) == 0; \\ Michel Marcus, Mar 03 2021

a(n) = A000720(A341804(n)) - 1.

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).

Original entry on oeis.org

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

Rémy Sigrist, Apr 14 2021

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.

			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.

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

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

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)) }

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.

A240673 Triangle read by rows: T(n, k) = v(n, k)*((1/v(n, k)) mod prime(k)), where v(n, k) = (Product_{j=1..n} prime(j))/prime(k), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 3, 4, 15, 10, 6, 105, 70, 126, 120, 1155, 1540, 1386, 330, 210, 15015, 20020, 6006, 25740, 16380, 6930, 255255, 170170, 306306, 145860, 46410, 157080, 450450, 4849845, 3233230, 3879876, 8314020, 6172530, 3730650, 9129120, 9189180

Offset: 1

Michel Marcus, Apr 10 2014

These coefficients have the following property: for any number j in 0..primorial(n)-1, j = Sum_{i=1..n} T(n,k)*(j mod prime(i)) mod primorial(n). For example, with the first 3 primes (2, 3, and 5) and j=47, j is [47 mod 2, 47 mod 3, 47 mod 5] = [1, 2, 2], and 15*1 + 10*2 + 6*2 = 15 + 20 + 12 = 47.

			Triangle starts:
     1;
     3,    4;
    15,   10,    6;
   105,   70,  126,  120;
  1155, 1540, 1386,  330,  210;

Matthias Schmitt, A function to calculate all relative prime numbers up to the product of the first n primes, arXiv:1404.0706 [math.NT] (see Example table on page 9).
Ramin Zahedi, On algebraic structure of the set of prime numbers, arXiv:1209.3165 [math.GM], 2012.
Ramin Zahedi, On a Deterministic Property of the Category of k-th Numbers: A Deterministic Structure Based on a Linear Function for Redefining the k-th Numbers in Certain Intervals, arXiv:1408.1888 [math.GM], 2014.

Cf. A000040 (primes), A002110 (primorials), A070826 (first column), A079276.

T := proc(n, k)
v := mul(ithprime(j), j=1..n)/ithprime(k);
v * ((1/v) mod ithprime(k)) end:
seq(print(seq(T(n,k), k=1..n)), n=1..7); # Peter Luschny, Apr 12 2014

lift[Rational[1, n_], p_] := Module[{m}, m /. Solve[n*m == 1, m, Modulus -> p][[1]]]; lift[1, 2] = 1;
v[n_, k_] := Product[Prime[j], {j, 1, n}]/Prime[k];
T[n_, k_] := v[n, k]*lift[1/v[n, k], Prime[k]];
Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 30 2017 *)

T(n, k) = {my(val = prod(j=1, n, prime(j))/prime(k)); val * lift(1/Mod(val, prime(k)));}

T(n, k) = v(n, k) * ((1/v(n, k)) mod prime(k)), where v(n, k) = Product_{j=1..n} prime(j)/prime(k).

T(n, n) = A002110(n-1) * A079276(n). - Peter Luschny, Apr 12 2014

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A341805 Numbers k such that (product of first k primes)-1 is divisible by the (k+1)-th prime.

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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).

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A240673 Triangle read by rows: T(n, k) = v(n, k)*((1/v(n, k)) mod prime(k)), where v(n, k) = (Product_{j=1..n} prime(j))/prime(k), n >= 1, 1 <= k <= n.

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