A073573 -id:A073573 - cp's OEIS frontend

A243583 Primes p for which p + 4 and p^3 + 4 are primes.

3, 7, 19, 79, 103, 109, 277, 379, 487, 967, 1489, 1663, 1867, 2857, 3019, 3253, 3613, 3697, 4003, 4783, 4969, 5413, 5437, 5503, 5569, 5647, 5923, 7477, 7669, 7687, 7699, 7789, 7933, 8233, 8779, 9007, 9319, 9547, 9739, 10597, 11257, 11467, 11593, 11827, 12037

Offset: 1

Abhiram R Devesh, Jun 09 2014

This is a subsequence of:
A023200: Primes p such that p + 4 is also prime.
A073573: Numbers n such that n^3 + 4 is prime.

			p = 3 is in this sequence because p + 4 = 7  (prime) and p^3 + 4 = 31 (prime).
p = 7 is in this sequence because p + 4 = 11  (prime) and p^3 + 4 = 347 (prime).

Abhiram R Devesh, Table of n, a(n) for n = 1..1000

Cf. A023200, A073573.

s=[]; forprime(p=2, 20000, if(isprime(p+4) && isprime(p^3+4), s=concat(s, p))); s \\ Colin Barker, Jun 11 2014

import sympy.ntheory as snt
n=2
while n>1:
    n1=n+4
    n2=((n**3)+4)
    ##Check if n1 and n2 are also primes.
    if snt.isprime(n1)== True and snt.isprime(n2)== True:
        print(n, " , " , n1, " , ", n2)
    n=snt.nextprime(n)

A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

Original entry on oeis.org

7, 5503, 21013, 301123, 303613, 420037, 469363, 679153, 771427, 991957, 999667, 1524763, 1707367, 2030653, 2333083, 2540563, 2552713, 2710933, 3009967, 3378103, 3441817, 3592213, 4419937, 4704613, 4840723, 5177797, 5691547, 6227587, 6275887, 6395677, 6595597, 6597163

Offset: 1

Zak Seidov, Aug 28 2014

nonn

For even k > 2, 4 + n^k is prime only for n = 1.
From Derek Orr, Aug 28 2014 (edited by Danny Rorabaugh, Apr 19 2015): (Start)
4+p^4 is composite for all primes p. For p = 2, 4+p^4 = 20 is composite. To prove it for odd primes, consider S(n) = 4+(2*n+1)^4. S(n) == 0 (mod 5) unless n == 2 (mod 5). If n == 2 (mod 5), then 2*n+1 == 0 (mod 5), which is only prime for n = 2; this gives p = 5 and 4+5^4 = 629 is composite. For other odd primes p, 4+p^4 is greater than 5 and divisible by 5.
4+p^(4*m) is also composite for any prime p and integer m > 0. For each m, the proof is the same as above.
(End)
All terms are == {3,7} (mod 10). - Zak Seidov, Aug 29 2014

			From _K. D. Bajpai_, Jan 20 2015: (Start)
a(2) = 5503:
4 + 5503 = 5507;
4 + 5503^2 = 30283013;
4 + 5503^3 = 166647398531;
4 + 5503^5 = 5046584669419727747;
all five are prime.
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007591, A073573, A125260, A172367.
Primes p such that 4+p^7, 4+p^9 and 4+p^11 are also prime is A253937. - K. D. Bajpai, Jan 20 2015
The subsequence with 4+p^7 also prime is A246562. - Danny Rorabaugh, Apr 19 2015

[p: p in PrimesUpTo(2*10^7) | IsPrime(4+p) and IsPrime(4+p^2) and IsPrime(4+p^3) and IsPrime(4+p^5)]; // Vincenzo Librandi, Apr 19 2015

k=4; Select[Prime[Range[1,500000]], PrimeQ[k+#]&&PrimeQ[k+#^2] &&PrimeQ[k+#^3] &&PrimeQ[k+#^5]&]  (*K. D. Bajpai, Jan 20 2015 *)

for(n=1, 6000000, if(isprime(n) && isprime(4+n) && isprime(4+n^2) && isprime(4+n^3) && isprime(4+n^5), print1(n, ", "))) \\ Colin Barker, Aug 28 2014

p=7; forprime(q=11, 1e8, if(q-p==4 && isprime(4+p^2) && isprime(4+p^3) && isprime(4+p^5), print1(p, ", ")); p=q) \\ Charles R Greathouse IV, Aug 28 2014

from sympy import prime, isprime
A246519_list = [p for p in (prime(n) for n in range(1,10**5)) if all([isprime(4+p**z) for z in (1,2,3,5)])]
# Chai Wah Wu, Sep 08 2014

A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

Original entry on oeis.org

7, 469363, 2552713, 3378103, 6595597, 6629683, 39837517, 46024063, 46167307, 97371007, 97629403, 105528217, 136983307, 169483033, 202953613, 213792193, 216520987, 216738043, 221705647, 304033927, 317502193, 359133553

Offset: 1

Zak Seidov, Aug 29 2014

nonn

All terms are == {3, 7} mod 10. Subsequence of A246519.

Zak Seidov, Table of n, a(n) for n = 1..60

Cf. A007591, A073573, A125260, A172367, A246519.

Select[Prime[Range[193*10^5]],AllTrue[#^{1,2,3,5,7}+4,PrimeQ]&] (* Harvey P. Dale, Sep 07 2024 *)

forprime(p=1,10^9,if(ispseudoprime(4+p) && ispseudoprime(4+p^2) && ispseudoprime(4+p^3) && ispseudoprime(4+p^5) && ispseudoprime(4+p^7), print1(p,", "))) \\ Derek Orr, Aug 30 2014

A201307 Primes of the form k^3+4.

Original entry on oeis.org

5, 31, 347, 733, 6863, 15629, 19687, 91129, 250051, 328513, 493043, 614129, 658507, 970303, 1092731, 1295033, 1520879, 1601617, 2146693, 2352641, 3048629, 4826813, 5359379, 6128491, 7414879, 8869747, 12977879, 21253937, 21717643

Offset: 1

Vincenzo Librandi, Nov 30 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A073573, A274077.

[a: n in [0..800] | IsPrime(a) where a is n^3+4];

Mathematica
```
Select[Table[n^3+4,{n,0,7000}],PrimeQ]
```

A243095 Least integer m > 1 such that 4 + m^n is prime or 1 if only 4 + 1^n is prime.

Original entry on oeis.org

3, 3, 3, 1, 7, 3, 7, 1, 3, 3, 9, 1, 33, 7, 9, 1, 43, 17, 27, 1, 9, 3, 7, 1, 55, 47, 285, 1, 27, 3, 39, 1, 43, 117, 163, 1, 63, 255, 15, 1, 87, 3, 43, 1, 187, 77, 37, 1, 105, 45, 25, 1, 99, 305, 79, 1, 3, 27, 903, 1, 127, 293, 255, 1, 27, 27, 435, 1, 207, 143, 127, 1, 117, 295, 1159, 1, 477

Offset: 1

Zak Seidov, Aug 29 2014

nonn

If n is a multiple of 4 then 4 + m^n is prime iff m = 1.

4 + m^(4*x) = (m^(2*x)-2*m^x+2) * (m^(2*x)+2*m^x+2). - Jens Kruse Andersen, Sep 02 2014

Zak Seidov and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 200 terms from Zak Seidov)

Cf. A007591, A073573, A125260, A172367, A246519.

a(n)=if(n%4==0,return(1));m=2;while(!ispseudoprime(4+m^n),m++);return(m)
vector(100,n,a(n)) \\ Derek Orr, Aug 29 2014

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A243583 Primes p for which p + 4 and p^3 + 4 are primes.

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A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

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A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

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A201307 Primes of the form k^3+4.

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A243095 Least integer m > 1 such that 4 + m^n is prime or 1 if only 4 + 1^n is prime.

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