Arsen Vardanyan - cp's OEIS frontend

A387039 Numbers k such that (p_k#)*(p_(k-1)#)+1, or A228593(k)+1 is prime.

1, 2, 3, 4, 5, 6, 11, 12, 35, 617

Offset: 1

Arsen Vardanyan, Aug 14 2025

			4 is a term since (p_4#)(p_3#) + 1 = (7*5*3*2)(5*3*2) + 1 = 210*30 + 1 = 6301 is prime.

Cf. A000040, A002110, A034386, A228593, A006094.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
q:= k-> isprime(p(k)*p(k-1)+1):
select(q, [$1..50])[];  # Alois P. Heinz, Aug 14 2025

Position[Times @@@ Partition[FoldList[Times, 1, Prime@ Range[400]], 2, 1] + 1, ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Aug 23 2025 *)

isok(k) = isprime(vecprod(primes(k))*vecprod(primes(k-1))+1);

a(10) from Michael S. Branicky, Aug 14 2025

A377248 Numbers k such that 8191 * 2^k + 1 is prime.

Original entry on oeis.org

12, 20, 412, 712, 2092, 4704, 10176, 33396, 41124, 105604, 139780, 142924

Offset: 1

Arsen Vardanyan, Oct 21 2024

8191 is the 5th Mersenne prime: 8191 = 2^13 - 1 (a term of A000668).

			12 is a term, because 8191 * 2^12 + 1 = 8191 * 4096 + 1 = 33550337 is prime. (also a term of A061644).

Cf. A000668, A002253, A002254, A001771, A061644.

PARI
```
is(k) = isprime(8191 * 2^k + 1);
```

a(8)-a(9) from Hugo Pfoertner, Oct 21 2024

a(10)-a(12) from Michael S. Branicky, Nov 05 2024

A375777 Numbers k such that k - rad(k) - 1 is prime, where rad(k) is A007947(k).

Original entry on oeis.org

8, 9, 12, 16, 18, 24, 25, 27, 28, 32, 36, 40, 45, 48, 49, 54, 56, 60, 63, 64, 75, 76, 84, 90, 96, 98, 100, 108, 112, 120, 121, 124, 126, 136, 144, 148, 152, 153, 160, 171, 175, 180, 184, 189, 196, 198, 204, 207, 208, 220, 228, 232, 243, 250, 261, 264, 270, 276

Offset: 1

Arsen Vardanyan, Aug 27 2024

nonn

			12 is a term, because 12 - rad(12) - 1 = 12 - (2 * 3) - 1 = 12 - 6 - 1 = 5 is prime.

Cf. A007947, A097379, A359213, A374593.

Select[Range[300], PrimeQ[# - Times @@ FactorInteger[#][[;; , 1]] - 1] &] (* Amiram Eldar, Aug 27 2024 *)

isok(k) = isprime(k - (vecprod(factor(k)[, 1])) - 1);

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

A374593 Numbers k such that k - rad(k) + 1 is prime, where rad(k) is the radical A007947(k).

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 32, 36, 40, 44, 45, 48, 49, 50, 56, 60, 63, 72, 75, 80, 81, 84, 88, 90, 92, 99, 104, 108, 116, 117, 128, 132, 136, 140, 144, 147, 153, 156, 160, 162, 164, 168, 169, 180, 184, 200, 204, 207, 212, 216, 224, 225, 234, 240, 243, 245, 250

Offset: 1

Arsen Vardanyan, Aug 23 2024

nonn

Includes 4*p for p in A005384, 8*p for p in A007693, and 16*p for p in A228857. - Robert Israel, Jun 27 2025

			12 is a term, because 12 - rad(12) + 1 = 12 - (2*3) + 1 = 12 - 6 + 1 = 7 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005384, A007693, A007947, A097379, A228857, A359213.

rad:= n -> convert(numtheory:-factorset(n),`*`):
select(k -> isprime(k - rad(k)+1), [$1..1000]); # Robert Israel, Jun 27 2025

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n);Select[Range[250],PrimeQ[#-rad[#]+1]&] (* James C. McMahon, Sep 27 2024 *)

isok(k) = isprime(k - (factorback(factor(k)[, 1])) + 1);

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

Original entry on oeis.org

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

A359695 Numbers k such that 29^k - 2 is prime.

Original entry on oeis.org

2, 4, 8, 14, 42, 420, 1344

Offset: 1

Arsen Vardanyan, Mar 07 2023

a(8) > 10^4, if it exists. - Amiram Eldar, Mar 10 2023
All terms in this sequence are even. - Yifan Xie, Mar 12 2023
a(8) > 5*10^4, if it exists. - Michael S. Branicky, Sep 14 2024

			4 is a term because 29^4 - 2 = 707279 is a prime number.

Cf. A087886 (29^k + 2 is prime).

Cf. A128460, A128459, A128457, A109076, A090669, A105772, A109080, (and similar others).

Select[Range[1400], PrimeQ[29^# - 2] &] (* Amiram Eldar, Mar 10 2023 *)

PARI
```
is(k) = isprime(29^k - 2);
```

A359180 Numbers k such that k!^2 / 2 + 1 is prime.

Original entry on oeis.org

2, 3, 6, 18, 19, 82, 1298, 3139, 3687, 4637

Offset: 1

Arsen Vardanyan, Dec 18 2022

			3!^2 / 2 + 1 = 6^2/2 + 1 = 19, a prime number, so 3 is a term.

Cf. A002981, A046029, A358805, A358878.

isok(k) = (k>1) && isprime(k!^2 / 2 + 1); \\ Michel Marcus, Jan 15 2023

a(7) from Michael S. Branicky, Dec 18 2022

a(8)-a(10) from Michael S. Branicky, Apr 10 2023

A359213 Numbers k such that rad(k) - 1 is prime.

Original entry on oeis.org

3, 6, 9, 12, 14, 18, 24, 27, 28, 30, 36, 38, 42, 48, 54, 56, 60, 62, 72, 74, 76, 81, 84, 90, 96, 98, 102, 108, 110, 112, 114, 120, 124, 126, 138, 144, 148, 150, 152, 158, 162, 168, 174, 180, 182, 192, 194, 196, 204, 216, 220, 224, 228, 230, 240, 243, 248, 252

Offset: 1

Arsen Vardanyan, Dec 21 2022

nonn

			rad(60) - 1 = 2*3*5 - 1 = 29, so 60 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007947, A097379.

Select[Range[250], PrimeQ[Times @@ FactorInteger[#][[;; , 1]] - 1] &] (* Amiram Eldar, Dec 21 2022 *)

isok(k) = isprime(factorback(factor(k)[, 1]) - 1); \\ Michel Marcus, Dec 22 2022

A358805 Numbers k such that k! + (k!/2) + 1 is prime.

Original entry on oeis.org

4, 5, 7, 11, 12, 14, 18, 28, 30, 62, 135, 153, 275, 584, 630, 1424, 1493, 4419, 8492, 10950

Offset: 1

Arsen Vardanyan, Dec 01 2022

Numbers k such that A070960(k)+1 is prime.

No more terms < 10000. - Vaclav Kotesovec, Dec 12 2022

Table of n, a(n) for n = 1..19
Arsen Vardanyan, A list of prime numbers

Cf. A070960, A358878.

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

a(18)-a(19) from Vaclav Kotesovec, Dec 09 2022

a(20) from Michael S. Branicky, Aug 02 2024

cp's OEIS Frontend

User: Arsen Vardanyan

A387039 Numbers k such that (p_k#)*(p_(k-1)#)+1, or A228593(k)+1 is prime.

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A377248 Numbers k such that 8191 * 2^k + 1 is prime.

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A375777 Numbers k such that k - rad(k) - 1 is prime, where rad(k) is A007947(k).

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

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A374593 Numbers k such that k - rad(k) + 1 is prime, where rad(k) is the radical A007947(k).

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

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A359695 Numbers k such that 29^k - 2 is prime.

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A359180 Numbers k such that k!^2 / 2 + 1 is prime.

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