A067200 -id:A067200 - cp's OEIS frontend

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

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

A144953 Primes of form n^3 + 2.

Original entry on oeis.org

2, 3, 29, 127, 24391, 91127, 250049, 274627, 328511, 357913, 571789, 1157627, 1442899, 1860869, 2146691, 2924209, 3581579, 5000213, 5177719, 6751271, 9129331, 9938377, 10503461, 12326393, 14348909, 14706127, 15438251, 18191449

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 26 2008

The Hardy-Littlewood conjecture K (p. 51) suggests that this sequence is infinite and gives an asymptotic estimate for the density of this sequence. - Charles R Greathouse IV, Jul 06 2010

Vincenzo Librandi and Robert Israel, Table of n, a(n) for n = 1..10000 (first 2900 terms from Vincenzo Librandi)
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

Cf. A067200.

[a: n in [0..800] | IsPrime(a) where a is n^3+2]; // Vincenzo Librandi, Nov 30 2011

N:= 10000: # number of terms desired
R[1]:= 2: count:= 1:
for n from 1 by 2 while count < N do
   p:= n^3+2;
   if isprime(p) then
     count:= count+1;
     R[count]:= p;
   end if
end do:
seq(R[n],n=1..N); # Robert Israel, Jan 29 2013

lst={};Do[s=n^3;If[PrimeQ[p=s+2],AppendTo[lst,p]],{n,6!}];lst
A144953={2};Do[If[PrimeQ[p=n^3+2],AppendTo[A144953,p]],{n,1,10^5,2}];A144953 (* Zak Seidov, Nov 05 2008 *)
Select[Table[n^3+2,{n,0,7000}],PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *)

for(n=0,1e3,if(isprime(k=n^3+2),print1(k","))) \\ Charles R Greathouse IV, Jul 06 2010

a(n) = A067200(n)^3 + 2. - Zak Seidov, Sep 16 2013

a(1)=2 from Zak Seidov, Nov 05 2008

Reference and index correction from Charles R Greathouse IV, Jul 06 2010

A125260 Numbers k such that k^5 + 4 is prime.

Original entry on oeis.org

1, 7, 9, 25, 39, 45, 73, 85, 99, 147, 163, 165, 169, 189, 213, 219, 223, 225, 249, 253, 259, 279, 333, 337, 385, 433, 457, 465, 469, 477, 499, 595, 639, 643, 655, 703, 709, 715, 729, 849, 853, 895, 969, 973, 979, 987, 1065, 1075, 1077, 1093, 1165, 1239, 1255

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..4431

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), this sequence (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)...

lst={};k=5;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 10^3}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[1300],PrimeQ[#^5+4]&] (* Harvey P. Dale, Oct 14 2019 *)

is(n)=isprime(n^5+4) \\ Charles R Greathouse IV, Feb 17 2017

A125259 Numbers k such that k^4 + 3 is prime.

Original entry on oeis.org

0, 2, 8, 16, 22, 26, 28, 34, 44, 62, 68, 76, 82, 92, 104, 110, 118, 128, 134, 166, 184, 202, 212, 266, 286, 296, 314, 328, 350, 356, 376, 406, 428, 436, 460, 470, 506, 520, 532, 562, 580, 638, 650, 652, 680, 692, 722, 734, 740, 778, 812, 820, 824, 862, 896, 908

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..5860

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), this sequence (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)...

Select[Range[0,1000],PrimeQ[#^4+3]&] (* Harvey P. Dale, Aug 02 2023 *)

isok(n, k=4) = isprime(n^k + k - 1); \\ Michel Marcus, Oct 11 2013

A125262 Numbers k such that k^7 + 6 is prime.

Original entry on oeis.org

1, 13, 17, 23, 61, 73, 77, 101, 137, 215, 221, 283, 307, 317, 361, 431, 457, 473, 481, 641, 731, 767, 817, 881, 985, 1015, 1061, 1145, 1235, 1283, 1333, 1337, 1343, 1531, 1693, 1711, 1817, 1847, 1853, 1867, 1903, 1963, 2057, 2093, 2113, 2161, 2201, 2363

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..2374

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259-A125265 (j=4..11).

lst={};k=7;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[2500],PrimeQ[#^7+6]&] (* Harvey P. Dale, May 05 2016 *)

is(n)=isprime(n^7+6) \\ Charles R Greathouse IV, Feb 17 2017

A125265 Numbers k such that k^11 + 10 is prime.

Original entry on oeis.org

1, 7, 19, 21, 33, 69, 153, 157, 193, 253, 379, 391, 439, 543, 549, 559, 579, 609, 879, 937, 939, 993, 1063, 1083, 1107, 1119, 1191, 1209, 1267, 1281, 1287, 1333, 1537, 1617, 1797, 1819, 1923, 1971, 1987, 1989, 2041, 2061, 2073, 2101, 2103, 2343, 2373

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10).

lst={};k=11;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[2500],PrimeQ[#^11+10]&] (* Harvey P. Dale, Jul 01 2015 *)

is(n)=isprime(n^11+10) \\ Charles R Greathouse IV, Feb 17 2017

A125261 Numbers k such that k^6 + 5 is prime.

Original entry on oeis.org

0, 18, 24, 114, 204, 216, 222, 246, 276, 312, 318, 372, 384, 426, 438, 468, 498, 582, 618, 654, 822, 888, 948, 984, 1182, 1188, 1272, 1278, 1284, 1374, 1446, 1536, 1674, 1782, 1788, 1794, 1806, 1812, 1896, 2034, 2058, 2088, 2124, 2154, 2232, 2238, 2376

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..2254

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), this sequence(j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)..

lst={};k=6;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0,2500],PrimeQ[#^6+5]&] (* Harvey P. Dale, Aug 29 2012 *)

is(n)=isprime(n^6+5) \\ Charles R Greathouse IV, Feb 17 2017

A125263 Numbers k such that k^8 + 7 is prime.

Original entry on oeis.org

0, 2, 4, 10, 66, 68, 86, 88, 134, 146, 200, 216, 250, 276, 306, 310, 410, 422, 472, 492, 506, 516, 538, 548, 550, 594, 638, 716, 746, 758, 862, 888, 942, 954, 964, 982, 992, 998, 1000, 1016, 1020, 1034, 1108, 1164, 1192, 1234, 1338, 1342, 1350, 1374, 1390

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..4683

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), this sequence (j=8), A125264 (j=10), A125265 (j=11)...

lst={};k=8;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0,1500],PrimeQ[#^8+7]&] (* Harvey P. Dale, Sep 04 2024 *)

is(n)=isprime(n^8+7) \\ Charles R Greathouse IV, Feb 17 2017

A125264 Numbers k such that k^10 + 9 is prime.

Original entry on oeis.org

2, 8, 238, 310, 338, 442, 542, 688, 698, 872, 920, 1198, 1330, 1382, 1538, 1558, 1678, 1702, 1712, 1768, 1882, 2032, 2080, 2102, 2260, 2312, 2408, 2440, 2540, 2642, 3112, 3170, 3188, 3268, 3320, 3580, 3740, 3742, 3770, 3980, 4028, 4048, 4148, 4292, 4472

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..1524

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), this sequence (j=10), A125265 (j=11).

lst={};k=10;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[5000],PrimeQ[#^10+9]&] (* Harvey P. Dale, Nov 17 2014 *)

is(n)=isprime(n^10+9) \\ Charles R Greathouse IV, Feb 17 2017

A159829 a(n) is the smallest natural number m such that n^3+m^3+1^3 is prime.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 15, 2, 3, 2, 11, 10, 9, 2, 7, 14, 5, 4, 9, 2, 15, 2, 7, 16, 15, 8, 13, 2, 1, 10, 3, 4, 15, 2, 11, 10, 9, 2, 7, 6, 13, 22, 5, 2, 1, 6, 29, 10, 29, 10, 3, 2, 11, 12, 3, 8, 3, 2, 19, 6, 15, 8, 1, 2, 1, 18, 5, 2, 1, 18, 1, 12, 17, 14, 15, 26, 7, 6, 3, 2, 19, 12, 1, 18, 3, 8, 15, 2, 11, 6

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009

nonn

a(2k-1) is odd, a(2k) is even.
Exponent 2: There are infinitely many primes of the forms n^2+m^2 and n^2+m^2+1^2.
Exponent k>2: Are there infinitely many primes of the forms n^k+m^k and n^k+m^k+1^k?

			2^3+2^3+1=17 = A000040(7); a(2)=2.
7^3+15^3+1=3719 = A000040(519); a(7)=15.
21^3+15^3+1=18523 = A000040(2122), a(21)=15.

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999.
A. Weil, Number theory: an approach through history, Birkhäuser 1984.
David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A069003, A159828.

Cf. A067200 (when m=1).

A159829 := proc(n) for m from 1 do if isprime(n^3+m^3+1) then RETURN(m) ; fi; od: end: seq(A159829(n),n=1..120) ; # R. J. Mathar, Apr 28 2009

snn[n_]:=Module[{n3=n^3,m=1},While[!PrimeQ[n3+1+m^3],m++];m]; Array[ snn,100] (* Harvey P. Dale, Sep 04 2019 *)

a(n) = my(m=1); while (!isprime(n^3+m^3+1^3), m++); m; \\ Michel Marcus, Nov 07 2023

Corrected and extended by R. J. Mathar, Apr 28 2009

cp's OEIS Frontend

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

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A144953 Primes of form n^3 + 2.

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A125260 Numbers k such that k^5 + 4 is prime.

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A125259 Numbers k such that k^4 + 3 is prime.

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A125262 Numbers k such that k^7 + 6 is prime.

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A125265 Numbers k such that k^11 + 10 is prime.

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A125261 Numbers k such that k^6 + 5 is prime.

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A125263 Numbers k such that k^8 + 7 is prime.

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PARI

A125264 Numbers k such that k^10 + 9 is prime.