A048637 -id:A048637 - cp's OEIS frontend

A048636 Primes of the form prime^3 + 2.

29, 127, 24391, 357913, 571789, 1442899, 5177719, 18191449, 30080233, 73560061, 80062993, 118370773, 127263529, 131872231, 318611989, 344472103, 440711083, 461889919, 590589721, 756058033, 865523179, 1095912793, 1298596573, 1341919729, 1524845953, 1697936059

Offset: 1

Enoch Haga

The first terms in the intersection with A092402, i.e., of the form p + 2^3, are (18191449, 1341919729, 2588282119, 3532642669, 16445197009, ...). - M. F. Hasler, Jan 13 2025

			a(2) = 127 = 5^3 + 2 and 5 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A048637.

Cf. A092402 (primes of the form p + 8), A321891 (union of the two); A188764 (primes of the form (product of distinct primes^3) + 2).

select(isprime, [ithprime(i)^3+2$i=1..300])[];  # Alois P. Heinz, Jan 13 2025

lst={};Do[s=Prime[n]^3;If[PrimeQ[p=s+2], AppendTo[lst, p]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 26 2008 *)

forprime (p=2,1100,if(isprime(p^3+2),print1(p^3+2,", "))) \\ Hugo Pfoertner, Oct 30 2018

A240110 Primes p such that p+2 and p^3+2 are also prime.

Original entry on oeis.org

3, 5, 29, 71, 311, 419, 431, 1031, 1091, 1151, 1451, 1931, 2339, 3371, 3461, 4001, 4421, 4799, 5651, 6269, 6551, 6569, 6761, 6779, 6869, 7559, 7589, 8219, 9011, 9281, 10301, 11069, 11489, 11549, 12161, 12239, 12251, 12539, 14081, 15641, 17189, 18059, 18119, 18521

Offset: 1

K. D. Bajpai, Apr 01 2014

nonn

All the terms in the sequence, except a(1), are congruent to 2 mod 3.

K. D. Bajpai, Table of n, a(n) for n = 1..3166

Cf. A000040, A048637, A140454, A175234, A236687, A236688.

KD := proc() local a,b,d; a:=ithprime(n);  b:=a+2;  d:=a^3+2;  if isprime(b)and isprime(d) then RETURN (a);  fi;  end:  seq(KD(), n=1..10000);

Select[Prime[Range[2200]],AllTrue[{#+2,#^3+2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 14 2017 *)

s=[]; forprime(p=2, 20000, if(isprime(p+2) && isprime(p^3+2), s=concat(s, p))); s \\ Colin Barker, Apr 01 2014

A240126 Primes p such that p - 2 and p^3 - 2 are also prime.

Original entry on oeis.org

19, 31, 109, 151, 241, 619, 859, 1489, 1951, 2131, 2791, 2971, 3559, 4129, 4651, 4789, 4801, 5659, 6661, 6781, 7591, 8221, 8629, 8821, 8971, 9241, 9721, 9931, 10891, 11971, 12109, 12541, 13831, 14011, 15271, 15289, 15331, 16831, 17029, 17419, 17839, 17989, 18121, 18541, 20149, 20899, 21019

Offset: 1

K. D. Bajpai, Apr 01 2014

nonn

All the terms in the sequence are congruent to 1 mod 3.

			19 is in the sequence because 19 is a prime: 19 - 2 = 17 and 19^3 - 2 = 6857 are also prime.
151 is in the sequence because 151 is a prime: 151 - 2 = 149 and 151^3 - 2 = 3442949 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..3085

Intersection of A006512 and A178251.

Cf. A048637, A140454, A175234, A236687, A236688, A240110.

KD := proc() local a,b,d; a:=ithprime(n); b:=a-2; d:=a^3-2;  if isprime(b)and isprime(d) then RETURN (a); fi; end: seq(KD(), n=1..10000);

Select[Prime[Range[2000]], PrimeQ[# - 2] && PrimeQ[#^3 - 2] &]

s=[]; forprime(p=2, 22000, if(isprime(p-2) && isprime(p^3-2), s=concat(s, p))); s \\ Colin Barker, Apr 02 2014

A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 29, 71, 83, 113, 173, 263, 311, 419, 431, 491, 503, 509, 683, 701, 761, 773, 839, 911, 953, 1031, 1091, 1103, 1151, 1193, 1259, 1283, 1373, 1451, 1523, 1583, 1601, 1733, 1823, 1889, 1931, 2099, 2153, 2213, 2273, 2339, 2351, 2441, 2531, 2543

Offset: 1

Jonathan Vos Post, Aug 16 2007

The corresponding near-cube primes are A132282. Analog of near-square primes. After a(1) = 0, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.

			a(1) = 0 because 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1 because 1^3 + 2 = 5 is prime and 1 is noncomposite.
a(3) = 3 because 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5 because 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29 because 29^3 + 2 = 24391 is prime.
45 is not in the sequence because, although 45^3 + 2 = 91127 is prime, 45 is not prime.
63 is not in the sequence because, although 63^3 + 2 = 250049 is prime, 63 is not prime.
65 is not in the sequence because, although 65^3 + 2 = 274627 is prime, 65 is not prime.
a(6) = 71 because 71^3 + 2 = 357913 is prime.
a(7) = 83 because 83^3 + 2 = 571789 is prime.
a(8) = 113 because 113^3 + 2 = 1442899 is prime.
123 is not in the sequence because, although 123^3 + 2 = 1860869 is prime, 123 is not prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error, arXiv:0706.1497 [math.NT], 2007.

Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132282.

{p in A000040 such that A067200(p) = A084380(p) = p^3 + 2 is in A000040}.

Union of {0,1} and A048637. - R. J. Mathar, Oct 18 2007

More terms from R. J. Mathar, Oct 18 2007

A261536 Primes p such that p^5 + 2 is also prime.

Original entry on oeis.org

11, 149, 179, 197, 281, 317, 389, 401, 419, 491, 509, 587, 977, 1019, 1217, 1289, 1367, 1499, 1607, 1637, 2039, 2111, 2339, 2459, 2609, 2801, 2897, 3119, 3221, 3359, 3701, 3767, 3917, 4451, 4517, 4871, 5237, 5531, 5717, 5879, 5927, 6197, 6311, 6959, 7151

Offset: 1

Vincenzo Librandi, Aug 24 2015

Subsequence of primes of A216976. - Michel Marcus, Aug 24 2015

All terms == 5 (mod 6). - Robert Israel, Sep 22 2019

			11^5 + 2 = 161053 is a prime.

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

Cf. primes p such that p^k+2 is also prime: A001359 (k=1), A048637 (k=3), this sequence (k=5), A261537 (k=7), A261538 (k=9).

Cf. A000040.

[p: p in PrimesUpTo(12000) | IsPrime(p^5+2)];

filter:= proc(p) isprime(p) and isprime(p^5+2) end proc:
select(filter, [seq(i,i=5..10000,6)]); # Robert Israel, Sep 22 2019

Select[Prime[Range[1000]], PrimeQ[#^5 + 2] &]

A242326 Primes p for which p + 2, p^3 + 2 and p^5 + 2 are prime.

Original entry on oeis.org

419, 2339, 14081, 45821, 46349, 51419, 56039, 68489, 70379, 108191, 112601, 115319, 131891, 132749, 256391, 267611, 278879, 314159, 328511, 342449, 361001, 385139, 424841, 433259, 470651, 489689, 519371, 573761, 664691, 691181, 694271

Offset: 1

Abhiram R Devesh, May 10 2014

nonn

Subsequence of A001359 and A048637.
All the terms in the sequence are congruent to 2 mod 3. This sequence is a subsequence of A240110.
Also, congruent to (11, 29) mod 30. - Zak Seidov, May 18 2014
Also, subsequence of A216976. - Michel Marcus, May 18 2014

			419 is in the sequence because
p = 419 (prime),
p + 2 = 421 (prime),
p^3 + 2 = 73560061 (prime), and
p^5 + 2 = 12914277518101 (prime).

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

Cf. A001359, A048637.

[p: p in PrimesUpTo(10^6)| IsPrime(p+2) and IsPrime(p^3+2)and IsPrime(p^5+2)]; // Vincenzo Librandi, May 11 2014

Select[Prime[Range[10^5]], PrimeQ[# + 2]&& PrimeQ[#^3 + 2]&& PrimeQ[#^5 + 2] &] (* Vincenzo Librandi, May 11 2014 *)

A242327 Primes p for which (p^n) + 2 is prime for n = 1, 3, 5, and 7.

Original entry on oeis.org

132749, 1175411, 3940799, 5278571, 11047709, 12390251, 15118769, 21967241, 22234871, 26568929, 31809959, 32229341, 32969591, 35760551, 38704661, 43124831, 43991081, 49248971, 50227211, 51140861, 53221631, 55568171, 59446109, 63671651, 71109161, 76675589

Offset: 1

Abhiram R Devesh, May 10 2014

nonn

Subsequence of A001359 and A048637.

			p = 132749 (prime);
p + 2 = 132751 (prime);
p^3 + 2 = 2339342304585751 (prime);
p^5 + 2 = 41224584878413873150038751 (prime);
p^7 + 2 = 726471878470342746448722269536491751 (prime).

Abhiram R Devesh, Table of n, a(n) for n = 1..50

Cf. A001359, A006512, A048637.

isok(p) = isprime(p) && isprime(p+2) && isprime(p^3+2) && isprime(p^5+2) && isprime(p^7+2); \\ Michel Marcus, May 15 2014

import sympy
from sympy.ntheory import isprime, nextprime
n=2
while True:
    n1=n+2
    n2=n**3+2
    n3=n**5+2
    n4=n**7+2
    ##.Check if n1, n2, n3 and n4 are also primes
    if all(isprime(x) for x in [n1, n2, n3, n4]):
        print(n, ", ", n1, ", ", n2, ", ", n3, ", ", n4)
    n=nextprime(n)

def is_A242327(n):
    return is_prime(n) and all([is_prime(n^(2*k+1)+2) for k in range(4)])
filter(is_A242327, range(3940800)) # Peter Luschny, May 15 2014

cp's OEIS Frontend

A048636 Primes of the form prime^3 + 2.

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A240110 Primes p such that p+2 and p^3+2 are also prime.

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A240126 Primes p such that p - 2 and p^3 - 2 are also prime.

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A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

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A261536 Primes p such that p^5 + 2 is also prime.

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A242326 Primes p for which p + 2, p^3 + 2 and p^5 + 2 are prime.

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A242327 Primes p for which (p^n) + 2 is prime for n = 1, 3, 5, and 7.

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