A001912 -id:A001912 - cp's OEIS frontend

A002643 Numbers k such that (k^2 + k + 1)/19 is prime.

7, 11, 26, 45, 83, 125, 140, 182, 197, 201, 216, 239, 258, 311, 330, 353, 444, 467, 482, 486, 524, 539, 558, 600, 752, 771, 843, 881, 885, 923, 980, 999, 1071, 1113, 1170, 1223, 1337, 1356, 1470, 1664, 1835, 1869, 1911, 1949, 1968, 2021, 2078, 2120, 2192

Offset: 1

N. J. A. Sloane

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..1000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

Cf. A002640, A002641, A002642, A002644.

Select[Range[2500], PrimeQ[(#^2 + # + 1)/19] &] (* Vincenzo Librandi, Sep 25 2012 *)

is(n)=isprime((n^2+n+1)/19) \\ Charles R Greathouse IV, Jun 06 2017

More terms from Jon E. Schoenfield, May 06 2010

A055755 4n^2+1, 2n^2+1, 2n^2-1 are all prime.

Original entry on oeis.org

3, 42, 45, 102, 132, 153, 237, 297, 375, 468, 570, 990, 2085, 2478, 2712, 3240, 4743, 5382, 5517, 6828, 7962, 8970, 8982, 9033, 9570, 9612, 9747, 9813, 10692, 12363, 12453, 12468, 12750, 13902, 14763, 14925, 15750, 16365, 17118, 17688, 19527

Offset: 1

Harvey P. Dale, Jul 12 2000

			42 is included because 4*42^2+1, 2*42^2+1, 2*42^2-1 are all prime numbers.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A001912.

with(numtheory): for n from 1 to 50000 do if isprime(4*n^2+1) and isprime(2*n^2+1) and isprime(2*n^2-1) then printf(`%d,`,n) fi: od:

a={};Do[If[PrimeQ[4n^2+1] && PrimeQ[2n^2+1] && PrimeQ[2n^2-1], AppendTo[a,n]], {n,10000}]; a (* Peter J. C. Moses, Apr 02 2013 *)

More terms from James Sellers, Jul 13 2000

A248887 Primes p of the form 4m^2+1 such that q=4p^2+1 and r=4q^2+1 are prime.

Original entry on oeis.org

677, 6635777, 28132417, 156400037, 234518597, 381655297, 386751557, 403849217, 820020497, 1215498497, 1298449157, 1539463697, 1580539537, 1839037457, 2072616677, 2774550277, 2969814017, 6147500837, 6194319617, 6703351877, 6937890437

Offset: 1

Zak Seidov, Mar 05 2015

nonn

All terms == 7 mod 10. Subsequence of A121834.

Cf. A001912, A121326.

A293621 Numbers k such that (2k)^2 + 1 and (2k+2)^2 + 1 are both primes.

Original entry on oeis.org

1, 2, 7, 12, 27, 62, 102, 192, 232, 317, 322, 357, 547, 572, 587, 622, 637, 657, 687, 782, 807, 837, 842, 982, 1027, 1042, 1047, 1202, 1227, 1267, 1332, 1417, 1462, 1567, 1652, 1767, 1877, 1887, 2012, 2077, 2087, 2182, 2302, 2307, 2367, 2392, 2397, 2477, 2507

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite. He gives the 24 terms below 10^3.

Sierpiński noted that the only triple of consecutive primes of the form (2n)^2 + 1 are for n = 1 (i.e., 1 and 2 are the only consecutive terms in this sequence), since every triple of consecutive terms contains at least one term which is divisible by 5.

			1 is in the sequence since (2*1)^2 + 1 = 5 and (2*1+2)^2 + 1 = 17 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wacław Sierpiński, Remarque sur la distribution de nombres premiers, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Schinzel's hypothesis H.

Subsequence of A001912.

Cf. A002496, A002522, A005574, A053755, A096012.

Select[Range[10^4], AllTrue[{(2#)^2+1, (2#+2)^2+1}, PrimeQ] &]

isok(n) = isprime((2*n)^2 + 1) && isprime((2*n+2)^2 + 1); \\ Michel Marcus, Oct 13 2017

a(n) = A096012(n)/2. - Amiram Eldar, Feb 24 2020

A002647 Sextan primes: p = (x^6 + y^6)/(x^2 + y^2).

Original entry on oeis.org

13, 61, 73, 193, 241, 541, 601, 1021, 1801, 1873, 1933, 2221, 3121, 3361, 4993, 5521, 6481, 8461, 9181, 9901, 10993, 11113, 12241, 12541, 13633, 14173, 17761, 20593, 21433, 21661, 21841, 23773, 26113, 27901, 28393, 29101, 34141, 41161, 49201

Offset: 1

N. J. A. Sloane

nonn

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).

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

More terms from Klaus Brockhaus, Jun 03 2009

A006687 Duodecimal primes: p = (x^12 + y^12 )/(x^4 + y^4 ).

Original entry on oeis.org

241, 5521, 6481, 51361, 346561, 380881, 390001, 1678321, 4332721, 4654801, 5576881, 12707521, 39336721, 41432641, 42942001, 99990001, 167948881, 184970641, 197063761, 205598881

Offset: 1

N. J. A. Sloane

nonn

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

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

A115449 Numbers n such that 4*n^5 - 1 is prime.

Original entry on oeis.org

1, 2, 3, 8, 12, 23, 27, 42, 68, 75, 86, 96, 113, 117, 125, 135, 140, 146, 168, 182, 185, 188, 191, 198, 233, 245, 255, 267, 281, 287, 297, 306, 311, 318, 327, 360, 362, 366, 377, 390, 392, 395, 408, 416, 423, 432, 447, 456, 465, 486, 488, 497, 516, 531, 555

Offset: 1

Parthasarathy Nambi, Mar 08 2006

nonn

			If n=96 then (4*n^5 - 1) = 32614907903 (prime).

Cf. A115104, A001912.

Select[Range[500], PrimeQ[4*#^5 - 1] &] (* Stefan Steinerberger, Mar 09 2006 *)

for(i=1,2000,if(isprime(4*i^5-1),print1(i,","))) \\ Matthew Conroy, Mar 12 2006

for(i=1,2000,if(isprime(4*i^5-1),print1(i,","))) \\ Matthew Conroy, Mar 12 2006

More terms from Stefan Steinerberger, Zak Seidov and Matthew Conroy, Mar 12 2006

More terms from Matthew Conroy, Mar 12 2006

A116947 Numbers k such that 4*k^6 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 17, 20, 22, 28, 30, 40, 45, 67, 68, 70, 75, 82, 85, 87, 88, 95, 108, 123, 125, 140, 150, 153, 163, 172, 190, 197, 200, 210, 217, 220, 223, 232, 237, 248, 268, 270, 282, 283, 287, 303, 310, 320, 333, 340, 358, 367, 403, 405, 407, 423, 438, 445, 447

Offset: 1

Parthasarathy Nambi, Apr 03 2006

			If k=197 then (4*k^6 + 1) is a prime with 15 digits.

Cf. A115104, A001912, A115349.

Select[Range[500], PrimeQ[(4*#^6 + 1)] &] (* Stefan Steinerberger, Apr 06 2006 *)

is(n)=isprime(4*n^6+1) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Stefan Steinerberger, Apr 06 2006

A248892 Primes p of the form 4m^2+1 such that q=4p^2+1, r=4q^2+1 and s=4r^2+1 are all prime.

Original entry on oeis.org

1565891838737, 1985917919077, 2060476510097, 5590084720897, 39623323626437, 94860314619877, 114027286862737, 115071875848337, 117140013119377, 136739205150917, 246382184192357, 254109295929637, 265883157493777, 340055949647237, 378534223873937

Offset: 1

Zak Seidov, Mar 05 2015

nonn

Corresponding values of k: 625678,704613,717718,1182168,3147353,4869813,5339178,5363578,5411562,5846777,7848283,7970403,8152962,9220303,9727978.

Subsequence of A248887. Cf. A001912, A121326, A121834, A248887.

apQ[p_]:=Module[{q=4p^2+1,r},r=4q^2+1;AllTrue[{p,q,r,4r^2+1},PrimeQ]]; Select[ 4*Range[ 10^7]^2+1,apQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 28 2019 *)

A274779 Numbers whose square is the sum of two positive triangular numbers in exactly one way.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 12, 13, 18, 20, 27, 28, 33, 37, 42, 45, 47, 55, 58, 60, 62, 63, 65, 67, 73, 75, 78, 80, 85, 88, 90, 92, 102, 103, 105, 112, 115, 118, 120, 125, 128, 130, 132, 135, 140, 142, 150, 153, 157, 163, 170, 175, 192, 193, 198, 200, 203, 210, 215, 218, 220, 222

Offset: 1

Altug Alkan, Jul 06 2016

Obviously, A000217(n) + A000217(n+1) = n*(n+1)/2 + (n+1)*(n+2)/2 = (n+1)^2. So every square that is greater than 1 is the sum of two positive consecutive triangular numbers. This sequence focuses on the squares that have only this trivial solution.

For a related comment, see comments section of A001912.

			3 is a term because 3^2 is the sum of two positive triangular numbers in exactly 1 way that is: 3^2 = 3 + 6.

Cf. A000217, A000290, A001912, A230312, A274758.

nR[n_]:= (SquaresR[2, n]+Plus@@ Pick[{-4, 4}, IntegerQ/@ Sqrt[{n, n/2}]])/8 ; nTr[n_] := nR[8*n + 2] - Boole@ IntegerQ@ Sqrt[8*n + 1]; Select[Range[250], nTr[#^2]==1 &] (* Giovanni Resta, Jul 08 2016 *)

cp's OEIS Frontend

A002643 Numbers k such that (k^2 + k + 1)/19 is prime.

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A055755 4n^2+1, 2n^2+1, 2n^2-1 are all prime.

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A248887 Primes p of the form 4m^2+1 such that q=4p^2+1 and r=4q^2+1 are prime.

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A293621 Numbers k such that (2*k)^2 + 1 and (2*k+2)^2 + 1 are both primes.

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A002647 Sextan primes: p = (x^6 + y^6)/(x^2 + y^2).

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References

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Extensions

A006687 Duodecimal primes: p = (x^12 + y^12 )/(x^4 + y^4 ).

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A115449 Numbers n such that 4*n^5 - 1 is prime.

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Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A116947 Numbers k such that 4*k^6 + 1 is prime.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A248892 Primes p of the form 4m^2+1 such that q=4p^2+1, r=4q^2+1 and s=4r^2+1 are all prime.

Views

Author

Keywords

Comments

A293621 Numbers k such that (2k)^2 + 1 and (2k+2)^2 + 1 are both primes.