A028873 -id:A028873 - cp's OEIS frontend

A038599 Numbers k such that k^3 - 2 is prime.

9, 15, 19, 27, 31, 37, 67, 91, 99, 109, 121, 129, 135, 145, 151, 165, 187, 189, 201, 207, 211, 217, 241, 259, 265, 267, 277, 279, 289, 319, 355, 357, 367, 369, 387, 391, 411, 417, 427, 435, 439, 445, 457, 459, 477, 489, 511, 525, 549, 555, 561, 615, 619, 621

Offset: 1

Jeff Burch

nonn

			15^3 - 2 = 3373 is prime, so 15 is in the sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A038600, A028870, A028873.

a[n_]:=n^x-y;lst={};x=3;y=2;Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
Select[Range@ 640, PrimeQ[#^3 - 2] &] (* Michael De Vlieger, May 10 2016 *)

is(n)=isprime(n^3-2) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A038600(n)+2)^(1/3). - Zak Seidov, May 10 2016

Missed term, 207, and more terms added by Zak Seidov, Mar 14 2009

A108701 Values of n such that n^2-2 and n^2+2 are both prime.

Original entry on oeis.org

3, 9, 15, 21, 33, 117, 237, 273, 303, 309, 387, 429, 441, 447, 513, 561, 573, 609, 807, 897, 1035, 1071, 1113, 1143, 1233, 1239, 1311, 1563, 1611, 1617, 1737, 1749, 1827, 1839, 1953, 2133, 2211, 2283, 2589, 2715, 2721, 2955, 3081, 3093, 3453, 3549, 3555, 3621, 3723, 3807

Offset: 1

John L. Drost, Jun 19 2005

Since x^2 + 2 is divisible by 3 unless x is divisible by 3, all elements are 3 mod 6.

Intersection of A067201 and A028870. - Robert Israel, Sep 11 2014

			21 is on the list since 21^2 - 2 = 439 and 21^2 + 2 = 443 are primes.

David Wells, Prime Numbers, John Wiley and Sons, 2005, p. 219 (article:'Siamese primes')

Nathaniel Johnston, Table of n, a(n) for n = 1..8750
Raymond A. Beauregard and E. R. Suryanarayan, Square-plus-two primes, Mathematical Gazette 85(502) 90-1.

Subsequence of A016945.

Cf. A016945, A028870, A028873, A038599, A067201, A153974.

[n: n in [3..3600 by 6] | IsPrime(n^2-2) and IsPrime(n^2+2)];  // Bruno Berselli, Apr 15 2011

select(n -> isprime(n^2-2) and isprime(n^2+2), [seq(6*i+3,i=0..1000)]); # Robert Israel, Sep 11 2014

Select[Range[5000], PrimeQ[#^2 - 2] && PrimeQ[#^2 + 2] &] (* Alonso del Arte, Sep 11 2014 *)

is(n)=isprime(n^2-2)&&isprime(n^2+2) \\ Charles R Greathouse IV, Jul 02 2013

Terms corrected by Charles R Greathouse IV, Sep 11 2014

A153974 Numbers n such that n^3 - 3 is prime.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 26, 34, 38, 40, 74, 80, 106, 110, 116, 124, 136, 158, 178, 184, 190, 206, 224, 230, 238, 256, 274, 280, 316, 320, 338, 340, 386, 410, 428, 446, 458, 464, 470, 484, 496, 530, 544, 550, 556, 590, 626, 634, 644, 646, 674, 710, 718, 728, 730

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

2^3 - 3 = 5 is prime, 4^3 - 3 = 61 is prime, ...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A028870, A028873, A038599, A108701, A153975.

[n: n in [2..500] | IsPrime(n^3-3)]; // Vincenzo Librandi, Nov 26 2010

a[n_]:=n^x-y;lst={};x=3;y=3;Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,0,6!}];lst
Select[Range[2, 1000], PrimeQ[#^3 - 3] &] (* G. C. Greubel, Sep 01 2016 *)

is(n)=isprime(n^3-3) \\ Charles R Greathouse IV, Feb 17 2017

First two terms 0,1, removed by Zak Seidov, Mar 14 2009

A028874 Primes of form k^2 - 3.

Original entry on oeis.org

13, 61, 97, 193, 397, 673, 1021, 1153, 1597, 1933, 2113, 3361, 4093, 4621, 6397, 7393, 7741, 8461, 9601, 12097, 12541, 13921, 15373, 16381, 18493, 19597, 20161, 21313, 26893, 29581, 36097, 37633, 40801, 42433, 43261, 47521, 48397

Offset: 1

Patrick De Geest

nonn

Also primes equal to the product of two consecutive odd numbers (A000466) minus 2. - Giovanni Teofilatto, Feb 11 2010

All terms are of the form 6m + 1. - Zak Seidov, May 01 2014

			61 is prime and equal to 8^2 - 3, so it is in the sequence.
67 is prime but it's 8^2 + 3 = 9^2 - 14, so it is not in the sequence.
9^2 - 3 = 78 but it's composite, so it's not in the sequence either.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
R. J. Mathar, Solutions to the exponential Diophantine 1 + p_1^x + p_2^y + p_3^z = w^2 for distinct primes p_1, p_2, p_3, 2022.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A002476 (Primes of form 6m + 1), A028871, A028872, A028873.

Primes terms in A082109. Subsequence of A068228. - Klaus Purath, Jan 09 2023

[a: n in [2..300] | IsPrime(a) where a is n^2-3 ]; // Vincenzo Librandi, Nov 08 2014

Select[Range[2, 250]^2 - 3, PrimeQ] (* Harvey P. Dale, Aug 07 2013 *)
Select[Table[n^2 - 3, {n, 2, 300}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *)

select(isprime, vector(100,n,n^2-3)) \\ Charles R Greathouse IV, Nov 19 2014

A028872 INTERSECT A000040. - Klaus Purath, Dec 07 2020

a(n) = A028873(n)^2 - 3. - Amiram Eldar, Mar 01 2025

A153975 Values of n such that n^2-3 and n^2+3 are both prime.

Original entry on oeis.org

4, 8, 10, 14, 64, 92, 112, 140, 146, 172, 218, 298, 304, 322, 326, 340, 350, 356, 416, 440, 470, 508, 554, 560, 580, 626, 634, 652, 668, 686, 694, 704, 728, 736, 746, 770, 806, 818, 868, 892, 920, 1054, 1082, 1102, 1130, 1156, 1196, 1256, 1264, 1378, 1418

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Intersection of A028873 and A049422. - Zak Seidov, Oct 12 2014

			4^2 - 3 = 13 and 4^2 + 3 = 19 are both primes, so 4 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A028870, A028873, A038599, A049422, A108701, A153974.

[n: n in [1..1400] | IsPrime(n^2-3) and IsPrime(n^2+3)]; // Vincenzo Librandi, Oct 12 2014

Select[Range[1500], PrimeQ[#^2 - 3] && PrimeQ[#^2 + 3] &] (* Vincenzo Librandi, Oct 12 2014 *)

is(n) = isprime(n^2-3) && isprime(n^2+3); \\ Altug Alkan, Sep 01 2016

Incorrect term 0 removed and Mma edited by Zak Seidov, Oct 12 2014

A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

Original entry on oeis.org

1, 5, 2, 11, 6, 3, 19, 12, 8, 4, 29, 20, 15, 10, 7, 41, 30, 24, 18, 14, 9, 55, 42, 35, 28, 23, 17, 13, 71, 56, 48, 40, 34, 27, 22, 16, 89, 72, 63, 54, 47, 39, 33, 26, 21, 109, 90, 80, 70, 62, 53, 46, 38, 32, 25, 131, 110, 99, 88, 79, 69, 61, 52, 45, 37, 31, 155, 132, 120, 108

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Beginning at row 6, the columns obey a 3rd-order recurrence:
c(n)=c(n-1)+c(n-2)-c(n-3)+1.
Except for initial terms, the first seven rows are A028387, A002378, A005563, A028552, A008865, A014209, A028873, and the first column, A004652.

			Corner:
1....5...11...19
2....6...12...20
3....8...15...24
4...10...18...28
The double interspersion A161179 begins thus:
1....4....7...12...17...24
2....3....8...11...18...23
5....6...13...16...25...30
9...10...19...22...33...38
Expel the odd-numbered columns and then swap rows 1 and 2, leaving
3....11...23...39
4....12...24...40
6....16...30...48
10...22...38...58
Then replace each of those numbers by its rank when all the numbers are jointly ranked.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A163253, A163254, A163255, A163256, A163258.

Let S(n,k) denote the k-th term in the n-th row. Four cases:
S(1,k)=k^2+k-1
S(2,k)=k^2+k
if n>1 is odd, then S(n,k)=k^2+(n-1)k+(n-1)(n-3)/4
if n>2 is even, then S(n,k)= k^2+(n-1)k+n(n-4)/4.

A083022 Numbers n such that 4*n^2 - 3 is prime.

Original entry on oeis.org

2, 4, 5, 7, 10, 13, 16, 17, 20, 22, 23, 29, 32, 34, 40, 43, 44, 46, 49, 55, 56, 59, 62, 64, 68, 70, 71, 73, 82, 86, 95, 97, 101, 103, 104, 109, 110, 125, 127, 133, 134, 148, 149, 152, 155, 160, 161, 163, 164, 166, 170, 175, 178, 181, 185, 208, 209, 218, 220

Offset: 1

Hugo Pfoertner, May 31 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A028874.

[n: n in [1..300] | IsPrime(4*n^2 - 3)]; // Vincenzo Librandi, Oct 05 2012

Select[Range[300], PrimeQ[4 #^2 - 3] &] (* Vincenzo Librandi, Oct 05 2012 *)

is(n)=isprime(4*n^2-3) \\ Charles R Greathouse IV, May 22 2017

a(n) = A028873(n)/2.

A296507 Numbers m such that m^2 - 13 is a prime.

Original entry on oeis.org

4, 6, 12, 18, 24, 30, 36, 54, 72, 84, 90, 96, 102, 114, 120, 138, 168, 186, 198, 204, 210, 216, 228, 240, 276, 294, 318, 330, 354, 360, 372, 378, 402, 414, 438, 444, 456, 480, 498, 504, 588, 600, 612, 618, 630, 636, 666, 678, 690, 714, 720, 726, 732, 738, 762

Offset: 1

Zak Seidov, Dec 13 2017

nonn

All terms except 4 are divisible by 6. - Robert Israel, Dec 13 2017

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000

Cf. A028870, A028873, A028876, A028882, A090696.

select(n -> isprime(n^2-13), 2*[$2..10^4]); # Robert Israel, Dec 13 2017

Reap[m=4;Do[If[PrimeQ[m^2-13],Sow[m]];m=m+2,{1000}]][[2,1]]
Select[Range[800],PrimeQ[#^2-13]&] (* Harvey P. Dale, Mar 06 2023 *)

isok(n) = isprime(n^2-13); \\ Michel Marcus, Dec 14 2017

A114335 Numbers k such that k^2 + 1 and k^2 - 3 are both prime.

Original entry on oeis.org

4, 10, 14, 20, 26, 40, 110, 124, 146, 206, 250, 326, 340, 350, 436, 440, 470, 634, 686, 704, 920, 1004, 1054, 1060, 1124, 1156, 1246, 1276, 1294, 1316, 1376, 1420, 1550, 1570, 1654, 1664, 1784, 1966, 1970, 2026, 2116, 2210, 2380, 2516, 2594, 2654, 2780

Offset: 1

John L. Drost, Feb 07 2006

nonn

This is the intersection of A028873 and A005574; this sequence is also a subset of A090120 (n such that closest primes above and below n^2 differ by 4).

			a(2) = 10 since 10^2 + 1 = 101 and 10^2 - 3 = 97 are both prime.

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

Cf. A005574, A028873, A090120.

[n: n in [2..100000] |IsPrime(n^2+1) and IsPrime(n^2-3)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[3,3000],AllTrue[#^2+{1,-3},PrimeQ]&] (* Harvey P. Dale, Jul 25 2024 *)

isok(k) = isprime(k^2 + 1) && isprime(k^2 - 3); \\ Amiram Eldar, Mar 01 2025

A153262 Squares such that square+-3=primes.

Original entry on oeis.org

16, 64, 100, 196, 4096, 8464, 12544, 19600, 21316, 29584, 47524, 88804, 92416, 103684, 106276, 115600, 122500, 126736, 173056, 193600, 220900, 258064, 306916, 313600, 336400, 391876, 401956, 425104, 446224, 470596, 481636, 495616, 529984

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

nonn

16-3=13,16+3=19,primes; 64-3=61,64+3=67,primes;...

The squared members of the intersection of A028873 and A049422. [From R. J. Mathar, Jan 03 2009]

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

Cf. A144938

lst={};Do[p=n^2;If[PrimeQ[p-3]&&PrimeQ[p+3],AppendTo[lst,p]],{n,7!}];lst
Select[Range[750]^2,And@@PrimeQ[#+{3,-3}]&] (* Harvey P. Dale, Dec 19 2012 *)

cp's OEIS Frontend

A038599 Numbers k such that k^3 - 2 is prime.

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A108701 Values of n such that n^2-2 and n^2+2 are both prime.

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A153974 Numbers n such that n^3 - 3 is prime.

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A028874 Primes of form k^2 - 3.

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A153975 Values of n such that n^2-3 and n^2+3 are both prime.

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Magma

Mathematica

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A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

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Formula

A083022 Numbers n such that 4*n^2 - 3 is prime.

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