A002407 -id:A002407 - cp's OEIS frontend

A111251 Numbers k such that 3k^2 + 3k + 1 is prime.

1, 2, 3, 4, 6, 9, 10, 11, 13, 14, 17, 23, 24, 25, 27, 28, 30, 32, 34, 37, 38, 41, 42, 45, 48, 49, 52, 55, 58, 62, 63, 66, 67, 74, 80, 81, 86, 88, 90, 91, 93, 95, 105, 108, 119, 123, 125, 128, 129, 136, 140, 142, 147, 153, 156, 157, 158, 164, 165, 170, 171, 172, 175

Offset: 1

Parthasarathy Nambi, Oct 31 2005

nonn

That is, positive integers k such that (k+1)^3 - k^3 is prime.
The Hardy-Littlewood constant 1.68109913... of this polynomial is approximately half that of the well-known Euler polynomial A221712, i.e., in comparison, only about half as many prime numbers are produced asymptotically as with k^2 + k + 41. - Hugo Pfoertner, Feb 10 2020
The primes that are obtained are called cuban primes and are in A002407. - Bernard Schott, Feb 13 2020

			For k=52, 3*52^2 + 3*52 + 1 = 8269 is prime, so 52 is a term.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000 (terms 1..1965 from Pierre CAMI)

Cf. A221712, A002407 (resulting primes), A002504, A121259.

[k: k in [1..180] | IsPrime(3*k^2 + 3*k + 1)]; // Marius A. Burtea, Feb 10 2020

Select[Range[200],PrimeQ[3#^2+3#+1]&] (* Harvey P. Dale, May 29 2017 *)

for(n=0,250,if(isprime(3*n^2+3*n+1),print1(n,",")))

a(n) = floor(sqrt(A002407(n)/3)). - Rémi Guillaume, Oct 16 2023
a(n) = A002504(n) - 1. - Rémi Guillaume, Oct 21 2023
a(n) = (A121259(n) - 1)/2. - Rémi Guillaume, Dec 29 2023

Extended by Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 02 2005

A113478 Number of cuban primes less than 10^n.

Original entry on oeis.org

0, 1, 4, 11, 28, 64, 173, 438, 1200, 3325, 9289, 26494, 76483, 221530, 645685, 1895983, 5593440, 16578830, 49347768, 147402214, 441641536, 1326941536, 3996900895, 12066234206, 36501753353

Offset: 0

Eric W. Weisstein, Jan 10 2006

			7, 19, 37, 61, 127 are the first few cuban primes, so a(1)=1 and a(2)=4.

Eric Weisstein's World of Mathematics, Cuban Prime

Cf. A002407, A002504.

A113478(n)=sum(k=1,sqrt(10^n/3),isprime(3*k*(k+1)+1)) \\ M. F. Hasler, Nov 28 2007

a(15)-a(18) from Donovan Johnson, Feb 05 2010

a(19)-a(24) from Hiroaki Yamanouchi, Oct 08 2015

A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

Original entry on oeis.org

13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249, 129793, 139969, 142573, 147853, 169933

Offset: 1

N. J. A. Sloane

nonn

Primes p such that p = 1 + 3*m^2 for some integer m (A111051). - Michael Somos, Sep 15 2005

			193 is a term since 193 = (9^3 - 7^3)/(9 - 7) is a prime.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
Eric Weisstein's World of Mathematics, Cuban Prime.
Wikipedia, Cuban prime.

Cf. A002407, A111051 (values of m).

A subsequence of A007645.

[a: n in [0..400] | IsPrime(a) where a is 3*n^2+1]; // Vincenzo Librandi, Dec 02 2011

Select[Table[3n^2+1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

{a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( cMichael Somos, Sep 15 2005 */

a(n) = 3*A111051(n)^2 + 1. - Paul F. Marrero Romero, Nov 03 2023

Entry revised by N. J. A. Sloane, Jan 29 2013

A121259 Numbers k such that (3*k^2 + 1)/4 is prime.

Original entry on oeis.org

3, 5, 7, 9, 13, 19, 21, 23, 27, 29, 35, 47, 49, 51, 55, 57, 61, 65, 69, 75, 77, 83, 85, 91, 97, 99, 105, 111, 117, 125, 127, 133, 135, 149, 161, 163, 173, 177, 181, 183, 187, 191, 211, 217, 239, 247, 251, 257, 259, 273, 281, 285, 295, 307, 313, 315, 317, 329, 331, 341

Offset: 1

Zak Seidov, Aug 23 2006

			(3*5^2 + 1)/4 = 19 is the 2nd prime of this form, so a(2) = 5.
(3*13^2 + 1)/4 = 127 is the 5th prime of this form, so a(5) = 13.
(3*19^2 + 1)/4 = 271 is the 6th prime of this form, so a(6) = 19.

Michel Marcus, Table of n, a(n) for n = 1..8144

Cf. comment by Michael Somos in A002407.

Cf. A002504, A111251, A111051 (simpler variant).

Select[Range[400],PrimeQ[(3#^2+1)/4]&]  (* Harvey P. Dale, Mar 24 2011 *)

is(n) = my(p=(3*n^2+1)/4); (denominator(p)==1) && isprime(p); \\ Charles R Greathouse IV, Jun 13 2017; edited by Michel Marcus, Oct 12 2023

a(n) = sqrt((4*A002407(n) - 1)/3). [corrected by Rémi Guillaume, Dec 07 2023]
a(n) = 2*A002504(n) - 1. - Hugo Pfoertner, Oct 07 2023
a(n) = 2*A111251(n) + 1. - Rémi Guillaume, Dec 06 2023

A221794 Number of primes of the form (x+1)^3 - x^3 with x <= 10^n.

Original entry on oeis.org

1, 7, 42, 263, 1965, 15282, 126826

Offset: 0

Vladimir Pletser, Jan 25 2013

nonn

Cuban primes are primes that are the difference of two consecutive cubes, p = (x+1)^3 - x^3 (A002407). They are also primes of the form 3k(k+1) + 1, where values for k+1 are in A002504.

Ed Pegg, Jr., MathWorld: Cuban Prime
Wikipedia, Cuban prime

Cf. A002407, A002504, A003215, A113478 (number of cuban primes < 10^n).

A276261 Centered 21-gonal primes.

Original entry on oeis.org

127, 211, 757, 2521, 2857, 6301, 8527, 16381, 19867, 23689, 24697, 27847, 32341, 37171, 38431, 42337, 66361, 68041, 82237, 89839, 97777, 103951, 114661, 140071, 152461, 162751, 170689, 192781, 204331, 216217, 231547, 240997, 284131, 308827, 353557, 357421, 385057, 389089

Offset: 1

Ilya Gutkovskiy, Aug 26 2016

nonn

Primes of the form (21*k^2 + 21*k + 2)/2.

Numbers k such that (21*k^2 + 21*k + 2)/2 is prime: 3, 4, 8, 15, 16, 24, 28, 39, 43, 47, 48, 51, 55, 059, 60, 63, 79, 80, 88, 92, 96, 99, ...

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for sequences related to centered polygonal numbers

Cf. A000040, A069178.

Cf. similar sequences of the centered k-gonal primes: A125602 (k = 3), A027862 (k = 4), A145838 (k = 5), A002407 (k = 6), A144974 (k = 7), A090562 (k = 10), A262344 (k = 11), A262493 (k = 13), A264821 (k = 14), A264822 (k = 15), A264823 (k = 16), A264824 (k = 17), A264825 (k = 18), A264844 (k = 19), A264845 (k = 20), A201715 (k = 24).

Intersection[Table[(21 k^2 + 21 k + 2)/2, {k, 0, 1000}], Prime[Range[33000]]]

lista(nn) = for(n=1, nn, if(isprime(p=(21*n^2 + 21*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

A201477 Primes of the form 3n^2 + 4.

Original entry on oeis.org

7, 31, 79, 151, 367, 1087, 1327, 1879, 2887, 3271, 4111, 4567, 6079, 7207, 8431, 15991, 16879, 17791, 19687, 23767, 24847, 25951, 34351, 39679, 42487, 49927, 51487, 54679, 56311, 63079, 73951, 102679, 104911, 111751, 123631, 126079, 128551

Offset: 1

Vincenzo Librandi, Dec 02 2011

Equivalently, generalized cuban primes of the form (x^3-y^3)/(x-y) with x=y+4 (cf. A002407, A007645). - N. J. A. Sloane, Jan 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A002407. A subsequence of A007645. - N. J. A. Sloane, Jan 29 2013

Cf. A111052 (corresponding values of n). - Zak Seidov, Feb 04 2016

[a: n in [0..400] | IsPrime(a) where a is 3*n^2+4];

Mathematica
```
Select[Table[3n^2+4,{n,0,700}],PrimeQ]
```

lista(nn) = {for (n=0, nn, if (isprime(p=3*n^2 + 4), print1(p, ", ")););} \\ Michel Marcus, Feb 04 2016

a(n) = 4 + 3*A111052(n)^2. - Zak Seidov, Feb 04 2016

A160432 Primes of the form 310^(2n) + 3*10^n + 1.

Original entry on oeis.org

7, 331, 300030001, 3000000003000000001

Offset: 1

Giacomo Fecondo, May 13 2009

nonn

Primes of the form (x^3-y^3)/(x-y) with x = y+1 (which gives A002407) and also y=10^k for some k.
These prime numbers (differences of consecutive cubes: A002407), for k>0, have only three digits different from zero. The first is 3, the middle digit is 3 and the final digit is 1. The other 2(k-1) digits are value 0.
If k=6*i or k=6*i-1 the number is always divisible by 7. [Giacomo Fecondo, May 22 2010]

			a(1) = 7 = (10^0+1)^3 -(10^0)^3 , 2^3-1^3.
a(2) = 331 =(10^1+1)^3 -(10^1)^3, 11^3-10^3.
a(3) = 300030001 = (10^4+1)^3 - (10^4)^3, 10001^3-10000^3.
a(1)= 3t(t+1)+1 with t=10^0; a(2)= 3t(t+1)+1 with t=10^1; a(3)= 3t(t+1)+1 with t=10^4.
For k=102 (k=6*17) the number (10^102+1)^3-(10^102)^3 is divisible by 7; for k=101 (k=6*17-1) the number (10^101+1)^3-(10^101)^3 is divisible by 7. [_Giacomo Fecondo_, May 22 2010]

Cf. A002407, A003215.

[a: n in [0..30] | IsPrime(a) where a is 3*10^(2*n) + 3*10^n + 1]; // Vincenzo Librandi, Jan 28 2013

Select[Table[3*10^(2 n) + 3*10^n + 1, {n, 0, 1000}], PrimeQ] (* Vincenzo Librandi, Jan 28 2013 *)

A160432(n,print_all=0,Start=0,Limit=9e9)={for(k=Start,Limit,ispseudoprime(p=3*100^k+3*10^k+1) & !(print_all & print1(p",")) & !n-- & return(p))} \\ - M. F. Hasler, Jan 28 2013

New name from Vincenzo Librandi, Jan 28 2013

A221717 Non-cuban primes.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 277, 281, 283, 293

Offset: 1

N. J. A. Sloane, Jan 29 2013

nonn

Primes not in A002407.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002407, A007645, A003627.

nn = 10; c = Select[Table[3 x^2 + 3 x + 1, {x, nn}], PrimeQ[#] &]; Complement[Prime[Range[PrimePi[c[[-1]]]]], c] (* T. D. Noe, Jan 30 2013 *)

A221792 Number of n-digit cuban primes.

Original entry on oeis.org

1, 3, 7, 17, 36, 109, 265, 762, 2125, 5964, 17205, 49989, 145047, 424155, 1250298, 3697457, 10985390, 32768938, 98054446, 294239322, 885300000, 2669959359, 8069333311, 24435519147

Offset: 1

Vladimir Pletser, Jan 25 2013

Cf. A002407, A113478, A221793.

a(n) = A113478(n) - A113478(n-1). - Jens Kruse Andersen, Jul 14 2014

a(19)-a(24) added from A113478 by Andrew Howroyd, Jan 14 2020

cp's OEIS Frontend

A111251 Numbers k such that 3*k^2 + 3*k + 1 is prime.

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A113478 Number of cuban primes less than 10^n.

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A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

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A121259 Numbers k such that (3*k^2 + 1)/4 is prime.

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A221794 Number of primes of the form (x+1)^3 - x^3 with x <= 10^n.

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A276261 Centered 21-gonal primes.

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A201477 Primes of the form 3n^2 + 4.

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A111251 Numbers k such that 3k^2 + 3k + 1 is prime.

A160432 Primes of the form 310^(2n) + 3*10^n + 1.