A360825 -id:A360825 - cp's OEIS frontend

A360805 Nonnegative integers k such that k! mod nextprime(k) is larger than k.

0, 31, 120, 283, 293, 712, 2872, 3287, 5028, 5129, 7088, 9553, 13229, 14232, 14799, 15113, 20153, 20830, 23239, 30233, 31430, 31667, 34443, 40654, 44298, 50184, 78877, 105834, 115281, 125120, 164253, 192103, 201590, 227747, 239910, 241910, 282230, 322550, 374370

Offset: 1

Alois P. Heinz, Feb 22 2023

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..213

Cf. A000142, A007978, A151800, A360825.

q:= n-> is(n! mod nextprime(n)>n):
select(q, [$0..20000])[];

from itertools import count, islice
from functools import reduce
from sympy import nextprime
def A360805_gen(startvalue=0): # generator of terms >= startvalue
    n = max(startvalue,0)
    m = nextprime(n)
    while True:
        a = m-1
        klist = []
        for i in range(m-1,n,-1):
            a = a*pow(i,-1,m)%m
            if a>i-1:
                klist.append(i-1)
        yield from sorted(klist)
        n, m = m, nextprime(m)
A360805_list = list(islice(A360805_gen(),30)) # Chai Wah Wu, Feb 24 2023

{ k >= 0 : k! mod nextprime(k) > k }.

A360825(a(n)) > a(n).

A375277 a(n) = n! (mod nextprime(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 2, 5, 1, 10, 1, 12, 3, 8, 1, 16, 1, 18, 4, 11, 1, 22, 22, 6, 5, 14, 1, 28, 1, 30, 33, 20, 31, 18, 1, 36, 7, 20, 1, 40, 1, 42, 8, 23, 1, 46, 19, 11, 9, 26, 1, 52, 30, 27, 10, 29, 1, 58, 1, 60, 43, 53, 56, 33, 1, 66, 12, 35, 1, 70, 1, 72, 27, 23, 66, 39, 1, 78

Offset: 0

Robert G. Wilson v, Sep 18 2024

Same as A360825 except at n=3.
a(n) = 1 iff n+2 is prime (A040976).
a(n) = n iff n+1 is prime (A006093).
a(n) > n iff n is in A360805.

Dominic McCarty, Table of n, a(n) for n = 0..10000

Cf. A360825 (essentially the same).

Cf. A000040, A000142, A006093, A040976, A360805.

a[n_] := Mod[n!, NextPrime[n]]; Array[a, 79, 0](* for large n *) a[n_] := Block[{m = NextPrime@ n, k = p = 1}, While[k < n +1, p = Mod[p*k, m]; k++]; p]

from functools import reduce
from sympy import nextprime
def A375277(n): return ((p:=nextprime(n))-1)*pow(reduce(lambda i, j:i*j%p, range(n+1,p),1),-1,p)%p # Chai Wah Wu, Oct 18 2024

cp's OEIS Frontend

A360805 Nonnegative integers k such that k! mod nextprime(k) is larger than k.

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