A291504 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i is not a prime.

1, 1, 0, 1, 3, 8, 48, 206, 1838, 13336, 133764, 1081556, 11046816, 108196128, 1555323224, 16279258144, 289771660328, 3495882548784, 66923393467216, 942785369844048, 15625264115770992, 315553823251866304, 5974132307015712032, 104979988889030774848

Offset

0,5

Links

Table of n, a(n) for n=0..23.

Example

1             is not a prime,
1 + 3         is not a prime,
1 + 3 + 2     is not a prime,
1 + 3 + 2 + 4 is not a prime.
So [1, 3, 2, 4] satisfies all the conditions.
---------------------------------------------
a(1) = 1: [[1]];
a(3) = 1: [[1, 3, 2]];
a(4) = 3: [[1, 3, 2, 4], [1, 3, 4, 2], [4, 2, 3, 1]];
a(5) = 8: [[1, 3, 2, 4, 5], [1, 3, 4, 2, 5], [1, 5, 2, 4, 3], [1, 5, 4, 2, 3], [4, 2, 3, 1, 5], [4, 2, 3, 5, 1], [4, 5, 1, 2, 3], [4, 5, 3, 2, 1]].

Mathematica

Table[Count[Permutations@ Range@ n, _?(AllTrue[Accumulate@ #, ! PrimeQ@ # &] &)], {n, 0, 10}] (* Michael De Vlieger, Aug 26 2017 *)

Prog

(Python)
from functools import cache
from sympy import primerange
def a(n):
    @cache
    def c(t, s): # count satisfying permutations where items in t must be placed and sum is s
        if s in primes: return 0
        if len(t) == 0: return 1
        S = set(t)
        return sum(c(tuple(sorted(S-{i})), s+i) for i in t)
    primes = set(primerange(1, n*(n+1)//2+1))
    if n == 0 or sum(range(1, n+1)) in primes: return int(n == 0)
    return c(tuple(range(1, n+1)), 0)
print([a(n) for n in range(18)]) # Michael S. Branicky, May 02 2026

Crossrefs

Cf. A067957, A291355.

Sequence in context: A210373 A247033 A309578 * A034183 A003216 A366755

Adjacent sequences: A291501 A291502 A291503 * A291505 A291506 A291507

Keyword

nonn

Author

Seiichi Manyama, Aug 25 2017

Extensions

a(0), a(12)-a(23) from Alois P. Heinz, Aug 25 2017

Status

approved