A325057 Number of positive integers k <= prime(n)# so that (k mod p_1) < (k mod p_2) < ... < (k mod p_n).

1, 2, 3, 7, 19, 94, 381, 2217, 10248, 64082, 572741, 3590815, 33731134, 291308123, 1896596488, 14675287694, 147847569839, 1642854121867, 12717640104203, 134707566446733, 1285768348848054, 9334472487460317, 97284913917125312, 922382339920122509, 10370484766702974615

Offset: 0

Author

Bert Dobbelaere, Sep 04 2019

Keywords

• nonn

Comments

This sequence emerges during computation of A306582 and A306612.

Examples

			a(3) = 7:
  Solutions for k that have increasing remainders modulo the first 3 primes:
  k   modulo  2   3   5
  =====================
  22          0 < 1 < 2
  28          0 < 1 < 3
   4          0 < 1 < 4
   8          0 < 2 < 3
  14          0 < 2 < 4
  23          1 < 2 < 3
  29          1 < 2 < 4
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A002110, A306582, A306612.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-1, j-1), j=1..min(i, ithprime(n))))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=0..24);  # Alois P. Heinz, Jan 04 2023

Python

from sympy import prime
def f(k, r, n):
    if k==n: return prime(k)-r
    global cache ; args = (k, r)
    if args in cache: return cache[args]
    rv = f(k+1, r+1, n)
    if r < (prime(k)-1): rv += f(k, r+1, n)
    cache[args]=rv ; return rv
def A325057(n):
    global cache ; cache = {}
    return f(1, 0, n)

Extensions

Name edited and a(0)=1 prepended by Alois P. Heinz, Jan 04 2023