A325057 -id:A325057 - cp's OEIS frontend

A306582 a(n) is the least integer k such that the remainder of k modulo p is strictly increasing over the first n primes.

0, 2, 4, 34, 52, 194, 502, 1138, 4042, 5794, 5794, 62488, 798298, 5314448, 41592688, 483815692, 483815692, 5037219688, 18517814158, 18517814158, 19566774820732, 55249201504132, 1257253598786974, 6743244322196288, 24165921989926702, 24165921989926702, 5346711077171356252, 47449991406350138602, 278545375679341352084, 5604477496256287791854

Offset: 1

Charlie Neder, Jun 03 2019

If "strictly increasing" is replaced with "nondecreasing", this sequence becomes A000004.

Trivially, a(n) <= A002110(n)-2. Equality only holds for n = 0.

			  a(n) modulo 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
  ==== ==================================================
     0        0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0, ...
     2        0, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2, ...
     4        0, 1, 4, 4,  4,  4,  4,  4,  4,  4,  4, ...
    34        0, 1, 4, 6,  1,  8,  0, 15, 11,  5,  3, ...
    52        0, 1, 2, 3,  8,  0,  1, 14,  6, 23, 21, ...
   194        0, 2, 4, 5,  7, 12,  7,  4, 10, 20,  8, ...
   502        0, 1, 2, 5,  7,  8,  9,  8, 19,  9,  6, ...
  1138        0, 1, 3, 4,  5,  7, 16, 17, 11,  7, 22, ...
  4042        0, 1, 2, 3,  5, 12, 13, 14, 17, 11, 12, ...
  5794        0, 1, 4, 5,  8,  9, 14, 18, 21, 23, 28, ...

Cf. A000004, A002110, A306612, A325057.

isok(k, n) = {my(last = -1, cur); for (i=1, n, cur = k % prime(i); if (cur <= last, return (0)); last = cur;); return (1);}
a(n) = {my(k=0); while(!isok(k, n), k++); k;} \\ Michel Marcus, Jun 04 2019

from sympy import prime
def A306582(n):
    plist, rlist, x = [prime(i) for i in range(1,n+1)], [0]*n, 0
    while True:
        for i in range(n-1):
            if rlist[i] >= rlist[i+1]:
                break
        else:
            return x
        for i in range(n):
            rlist[i] = (rlist[i] + 1) % plist[i]
        x += 1 # Chai Wah Wu, Jun 15 2019

a(16)-a(20) from Daniel Suteu, Jun 03 2019
a(21)-a(23) from Giovanni Resta, Jun 16 2019
a(24)-a(27) from Bert Dobbelaere, Jun 22 2019
a(28)-a(30) from Bert Dobbelaere, Sep 05 2019

A354621 Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= prime(i).

Original entry on oeis.org

1, 2, 5, 19, 85, 586, 3583, 28568, 195449, 1666786, 18757980, 161386953, 1897428757, 20910643255, 186584844271, 1896239913403, 23753305611756, 322385257985845, 3291722491175736, 43011227141438328, 517673545204963277, 5056620552149902641, 65366993167319822971

Offset: 0

Alois P. Heinz, Jul 08 2022

nonn

The number of n-tuples of primes with p_{i-1} <= p_i <= prime(i) give A000108.

			a(0) = 1: ( ).
a(1) = 2: (1), (2).
a(2) = 5: (1,1), (1,2), (1,3), (2,2), (2,3).

Alois P. Heinz, Table of n, a(n) for n = 0..754

Cf. A000040, A000108, A325057.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-1, j), j=1..min(i, ithprime(n))))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=0..23);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)*
      (-1)^(n-j)*binomial(ithprime(j+1), n-j), j=0..n-1))
    end:
seq(a(n), n=0..23);

a[n_] := a[n] = If[n == 0, 1, -Sum[a[j]*(-1)^(n - j)* Binomial[Prime[j + 1], n - j], {j, 0, n - 1}]];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 28 2022, after second Maple program *)

a(n) = Sum_{j=0..n-1} a(j)*(-1)^(n+1-j)*binomial(prime(j+1),n-j) with a(0) = 1.

Sum_{n>=0} a(n)*x^n * (1-x)^prime(n+1) = 1.

cp's OEIS Frontend

A306582 a(n) is the least integer k such that the remainder of k modulo p is strictly increasing over the first n primes.

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A354621 Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= prime(i).

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