A359180 -id:A359180 - cp's OEIS frontend

A374901 Numbers k such that k!^2 + ((k - 1)!^2) + 1 is prime.

1, 3, 4, 6, 10, 11, 118, 271, 288, 441, 457, 2931, 5527, 6984, 9998, 10395, 13703

Offset: 1

Arsen Vardanyan, Jul 31 2024

a(18) > 15000 - Karl-Heinz Hofmann, Aug 23 2024

			4 is a term, because 4!^2 + 3!^2 + 1 = 576 + 36 + 1 = 613 is a prime number.

Cf. A000142, A055490, A358805, A358878, A359180, A080778, A243078, A242994

PARI
```
is(k) = isprime((k!^2)+((k-1)!)^2+1);
```

from itertools import count, islice
from sympy import isprime
def A374901_gen(): # generator of terms
    f = 1
    for k in count(1):
        if isprime((k**2+1)*f+1):
            yield k
        f *= k**2
A374901_list = list(islice(A374901_gen(),10)) # Chai Wah Wu, Oct 02 2024

a(12)-a(14) from Michael S. Branicky, Aug 01 2024

a(15)-a(17) from Karl-Heinz Hofmann, Aug 23 2024

A375310 Numbers k such that k!^2 + (k-1)!^2 - 1 is prime.

Original entry on oeis.org

14, 32, 58, 182, 240, 474, 824, 3018, 5977, 9088

Offset: 1

Arsen Vardanyan, Aug 11 2024

Contains no primes. - Robert Israel, Aug 12 2024
Moreover: If k is any prime, then k must divide the given formula and if k is not a prime the least factor that divides the formula is bigger than k and smaller than the square root of the result of the formula (if it exist, if not: k is a term). - Karl-Heinz Hofmann, Aug 20 2024
a(11) > 15000. -  Karl-Heinz Hofmann, Sep 08 2024

			14 is a term, because 14!^2 + 13!^2 - 1 = 7600054456551997440000 + 38775788043632640000 - 1 = 7638830244595630079999 is a prime number.

Cf. A374901.

Cf. A000142, A055490, A358805, A358878, A359180.

select(k -> isprime((k^2+1)*((k-1)!)^2-1), [$1..1000]); # Robert Israel, Aug 12 2024

PARI
```
is(k) = isprime(k!^2 + (k-1)!^2 - 1);
```

from itertools import count, islice
from sympy import isprime
def A375310_gen(): # generator of terms
    f = 1
    for k in count(1):
        if isprime((k**2+1)*f-1):
            yield k
        f *= k**2
A375310_list = list(islice(A375310_gen(),6)) # Chai Wah Wu, Oct 02 2024

a(8) from Hugo Pfoertner, Aug 13 2024
a(9) from Michael S. Branicky, Aug 14 2024
a(10) from Karl-Heinz Hofmann, Sep 08 2024

cp's OEIS Frontend

A374901 Numbers k such that k!^2 + ((k - 1)!^2) + 1 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Extensions