A066436 -id:A066436 - cp's OEIS frontend

A155923 Positive numbers y such that y^2 is of the form x^2+(x+17)^2 with integer x.

13, 17, 25, 53, 85, 137, 305, 493, 797, 1777, 2873, 4645, 10357, 16745, 27073, 60365, 97597, 157793, 351833, 568837, 919685, 2050633, 3315425, 5360317, 11951965, 19323713, 31242217, 69661157, 112626853, 182092985, 406014977, 656437405

Offset: 1

Klaus Brockhaus, Feb 09 2009

(-5,a(1)) and (A118120(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2. (Offset 1 is assumed for A118120.)
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (19+6*sqrt(2))/17 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (387+182*sqrt(2))/17^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, the x values are given by the sequence defined by: a(n)=6*a(n-3)-a(n-6)+2p with a(1)=0, a(2)=2m+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2*m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. [From Mohamed Bouhamida, Sep 09 2009]

			(-5,a(1)) = (-5,13) is a solution: (-5)^2+(-5+17)^2 = 25+144 = 169 = 13^2;
(A118120(1), a(2)) = (0, 17) is a solution: 0^2+(0+17)^2 = 289 = 17^2;
(A118120(2), a(3)) = (7, 25) is a solution: 7^2+(7+17)^2 = 49+576 = 625 = 25^2.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A118120, A156035 (decimal expansion of 3+2*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17), A157649 (decimal expansion of (387+182*sqrt(2))/17^2).

Cf. A156156 (first trisection), A156157 (second trisection), A156158 (third trisection).

LinearRecurrence[{0,0,6,0,0,-1},{13,17,25,53,85,137},50] (* Harvey P. Dale, Feb 11 2015 *)

{forstep(n=-5, 660000000, [1,3], if(issquare(2*n*(n+17)+289, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1) = 13, a(2) = 17, a(3) = 25, a(4) = 53, a(5) = 85, a(6) = 137.

G.f.: x*(1-x)*(13+30*x+55*x^2+30*x^3+13*x^4)/(1-6*x^3+x^6).

G.f. corrected, first and fourth comment and examples edited, cross-reference added by Klaus Brockhaus, Sep 22 2009

A091176 Numbers n such that prime(n) is of the form 2*k^2 - 1.

Original entry on oeis.org

4, 7, 11, 20, 25, 31, 46, 53, 68, 87, 106, 118, 152, 163, 190, 204, 247, 344, 377, 418, 436, 474, 492, 516, 558, 580, 647, 669, 713, 816, 894, 975, 1003, 1028, 1179, 1300, 1392, 1526, 1561, 1695, 1768, 1917, 1952, 2069, 2177, 2343, 2601, 2643, 2769, 2812

Offset: 1

Ray Chandler, Dec 25 2003

nonn

A066436 indexed by A000040.

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

Cf. A066049, A066436, A090697, A110558.

Flatten[Position[Prime[Range[3000]],?(IntegerQ[Surd[(#+1)/2,2]]&)]] (* _Harvey P. Dale, Jan 23 2013 *)

A000040(n) = A066436(n).

A129991 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

Original entry on oeis.org

0, 23, 620, 723, 840, 4223, 4820, 5499, 25200, 28679, 32636, 147459, 167736, 190799, 860036, 978219, 1112640, 5013239, 5702060, 6485523, 29219880, 33234623, 37800980, 170306523, 193706160, 220320839, 992619740, 1129002819, 1284124536, 5785412399, 6580311236

Offset: 1

Mohamed Bouhamida, Jun 14 2007

Also values x of Pythagorean triples (x, x+241, y).
Corresponding values y of solutions (x, y) are in A159565.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159565, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 23, 620, 723, 840, 4223, 4820}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+482*n+58081), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+482 for n > 6; a(1)=0, a(2)=23, a(3)=620, a(4)=723, a(5)=840, a(6)=4223.
G.f.: x*(23+597*x+103*x^2-21*x^3-199*x^4-21*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 241*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 16 2009

A130004 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.

Original entry on oeis.org

0, 31, 1204, 1347, 1504, 8151, 8980, 9891, 48600, 53431, 58740, 284347, 312504, 343447, 1658380, 1822491, 2002840, 9666831, 10623340, 11674491, 56343504, 61918447, 68045004, 328395091, 360888240, 396596431, 1914027940, 2103411891, 2311534480, 11155773447

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+449, y).
Corresponding values y of solutions (x, y) are in A159589.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159589, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).

I:=[0, 31, 1204, 1347, 1504, 8151, 8980]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) -Self(n-6) +Self(n-7): n in [1..30]]; // G. C. Greubel, May 08 2018

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 31, 1204, 1347, 1504, 8151, 8980}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+898*n+201601), print1(n, ",")))}

x='x+O('x^30); concat([0], Vec(x*(31+1173*x+143*x^2-29*x^3-391*x^4 -29*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 08 2018

a(n) = 6*a(n-3) -a(n-6) +898 for n > 6; a(1)=0, a(2)=31, a(3)=1204, a(4)=1347, a(5)=1504, a(6)=8151.
G.f.: x*(31+1173*x+143*x^2-29*x^3-391*x^4-29*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 449*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 17 2009

A130013 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

Original entry on oeis.org

0, 37, 1768, 1941, 2128, 11937, 12940, 14025, 71148, 76993, 83316, 416245, 450312, 487165, 2427616, 2626173, 2840968, 14150745, 15308020, 16559937, 82478148, 89223241, 96519948, 480719437, 520032720, 562561045, 2801839768, 3030974373

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+647, y).
Corresponding values y of solutions (x, y) are in A159641.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (649+36*sqrt(2))/647 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159641, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,37,1768,1941,2128,11937,12940},40] (* Harvey P. Dale, Jan 27 2025 *)

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1294 for n > 6; a(1)=0, a(2)=37, a(3)=1768, a(4)=1941, a(5)=2128, a(6)=11937.
G.f.: x*(37+1731*x+173*x^2-35*x^3-577*x^4-35*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 647*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 21 2009

A130017 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.

Original entry on oeis.org

0, 45, 2688, 2901, 3128, 18105, 19340, 20657, 107876, 115073, 122748, 631085, 673032, 717765, 3680568, 3925053, 4185776, 21454257, 22879220, 24398825, 125046908, 133352201, 142209108, 728829125, 777235920, 828857757, 4247929776

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+967, y).
Corresponding values y of solutions (x, y) are in A159701.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (969+44**sqrt(2))/967 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159701, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44**sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,45,2688,2901,3128,18105,19340},40] (* Harvey P. Dale, Nov 03 2013 *)

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1934 for n > 6; a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105.
G.f.: x*(45+2643*x+213*x^2-43*x^3-881*x^4-43*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 967*A001652(k) for k >= 0.
a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105, a(7)=19340, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Nov 03 2013

Edited and two terms added by Klaus Brockhaus, Apr 21 2009

A143829 Numbers n such that 10n^2 - 1 is prime.

Original entry on oeis.org

3, 6, 9, 12, 21, 30, 33, 36, 45, 48, 60, 69, 72, 75, 81, 87, 99, 108, 111, 114, 117, 120, 123, 126, 129, 153, 165, 168, 174, 177, 183, 201, 204, 207, 222, 234, 243, 252, 267, 279, 282, 285, 294, 303, 312, 315, 318, 339, 345, 348, 369, 378, 381, 384, 393, 396

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

A066436, A066049, A090686, A090684, A143826, A143827, A143828

p = 10; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, x]], {x, 1, 1000}]; a
Select[Range[500],PrimeQ[10#^2-1]&] (* Harvey P. Dale, Nov 11 2020 *)

is(n)=isprime(10*n^2-1) \\ Charles R Greathouse IV, Feb 17 2017

A090697 Numbers n such that n^2/2 - 1 is a prime.

Original entry on oeis.org

4, 6, 8, 12, 14, 16, 20, 22, 26, 30, 34, 36, 42, 44, 48, 50, 56, 68, 72, 76, 78, 82, 84, 86, 90, 92, 98, 100, 104, 112, 118, 124, 126, 128, 138, 146, 152, 160, 162, 170, 174, 182, 184, 190, 196, 204, 216, 218, 224, 226, 230, 236, 250, 252, 254, 264, 268, 274, 280

Offset: 1

Giovanni Teofilatto, Dec 20 2003

A066436 gives resulting primes p such that 2p+2 is square. - Ray Chandler, Dec 25 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

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

Cf. A066049, A066436, A091176, A110558.

Select[Range[2,300,2],PrimeQ[#^2/2-1]&] (* Harvey P. Dale, Apr 05 2014 *)

isok(n) = !(n % 2) && isprime(n^2/2 - 1); \\ Michel Marcus, Jul 23 2016

a(n) = 2*A066049(n) = A110558(n)/2. - Ray Chandler, Dec 25 2003

Corrected and extended by Ray Chandler, Dec 25 2003

A110558 Numbers n such that (n^2-8)/8 is prime.

Original entry on oeis.org

8, 12, 16, 24, 28, 32, 40, 44, 52, 60, 68, 72, 84, 88, 96, 100, 112, 136, 144, 152, 156, 164, 168, 172, 180, 184, 196, 200, 208, 224, 236, 248, 252, 256, 276, 292, 304, 320, 324, 340, 348, 364, 368, 380, 392, 408, 432, 436, 448, 452, 460, 472, 500, 504, 508

Offset: 1

Pierre CAMI, Sep 12 2005

nonn

These numbers need to be of the form 4*j then (16*j^2-8)/8 = 2*j^2-1.

A066436 gives resulting primes p such that 8p+8 is square. - Ray Chandler, Sep 15 2005

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A066049, A066436, A090697, A091176.

Select[Range[510],PrimeQ[(#^2-8)/8]&] (* Harvey P. Dale, Feb 03 2011 *)

is(n)=isprime((n^2-8)/8) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = 2*A090697(n) = 4*A066049(n). - Ray Chandler, Sep 15 2005

Extended by Ray Chandler, Sep 15 2005

A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

Original entry on oeis.org

0, 27, 888, 1011, 1148, 6027, 6740, 7535, 35948, 40103, 44736, 210335, 234552, 261555, 1226736, 1367883, 1525268, 7150755, 7973420, 8890727, 41678468, 46473311, 51819768, 242920727, 270867120, 302028555, 1415846568, 1578730083

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+337, y).
Corresponding values y of solutions (x, y) are in A159574.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (339+26*sqrt(2))/337 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159574, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576 (decimal expansion of (278307+179662*sqrt(2))/337^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,27,888,1011,1148,6027,6740},40] (* Harvey P. Dale, Feb 26 2015 *)

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+674*n+113569), print1(n, ",")))}

a(n)=6*a(n-3)-a(n-6)+674 for n > 6; a(1)=0, a(2)=27, a(3)=888, a(4)=1011, a(5)=1148, a(6)=6027.
G.f.: x*(27+861*x+123*x^2-25*x^3-287*x^4-25*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 337*A001652(k) for k >= 0.
a(0)=0, a(1)=27, a(2)=888, a(3)=1011, a(4)=1148, a(5)=6027, a(6)=6740, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Feb 26 2015

Edited and two terms added by Klaus Brockhaus, Apr 16 2009

cp's OEIS Frontend

A155923 Positive numbers y such that y^2 is of the form x^2+(x+17)^2 with integer x.

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A091176 Numbers n such that prime(n) is of the form 2*k^2 - 1.

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A129991 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

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A130004 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.

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A130013 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

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A130017 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.

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A143829 Numbers n such that 10n^2 - 1 is prime.

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