A066436 -id:A066436 - cp's OEIS frontend

A130005 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.

0, 35, 1568, 1731, 1908, 10595, 11540, 12567, 63156, 68663, 74648, 369495, 401592, 436475, 2154968, 2342043, 2545356, 12561467, 13651820, 14836815, 73214988, 79570031, 86476688, 426729615, 463769520, 504024467, 2487163856, 2703048243

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+577, y).
Corresponding values y of solutions (x, y) are in A159626.
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) = (579+34*sqrt(2))/577 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 for n mod 3 = 0.

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

Cf. A159626, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159627 (decimal expansion of (579+34*sqrt(2))/577), A159628 (decimal expansion of (855171+556990*sqrt(2))/577^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,35,1568,1731,1908,10595,11540},30] (* Harvey P. Dale, May 27 2018 *)

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+1154*n+332929), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1154 for n > 6; a(1)=0, a(2)=35, a(3)=1568, a(4)=1731, a(5)=1908, a(6)=10595.
G.f.: x*(35+1533*x+163*x^2-33*x^3-511*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 577*A001652(k) for k >= 0.

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

A130014 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

Original entry on oeis.org

0, 43, 2440, 2643, 2860, 16443, 17620, 18879, 97980, 104839, 112176, 573199, 613176, 655939, 3342976, 3575979, 3825220, 19486419, 20844460, 22297143, 113577300, 121492543, 129959400, 661979143, 708112560, 757461019, 3858299320

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+881, y).
Corresponding values y of solutions (x, y) are in A159690.
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) = (883+42*sqrt(2))/881 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 0.

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

Cf. A159690, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,43,2440,2643,2860,16443,17620},30] (* Harvey P. Dale, Aug 13 2015 *)

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1762*n+776161), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1762 for n > 6; a(1)=0, a(2)=43, a(3)=2440, a(4)=2643, a(5)=2860, a(6)=16443.
G.f.: x*(43+2397*x+203*x^2-41*x^3-799*x^4-41*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 881*A001652(k) for k >= 0.

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

A143830 Primes of the form 12*n^2-1.

Original entry on oeis.org

11, 47, 107, 191, 431, 587, 971, 1451, 2027, 2351, 2699, 3467, 4799, 5807, 6911, 7499, 8111, 8747, 10091, 10799, 14699, 15551, 16427, 17327, 18251, 25391, 27647, 36299, 41771, 44651, 55487, 57131, 62207, 67499, 71147, 74891, 80687, 92927, 99371

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Equals A089682 without the 2. [Sketch of proof: the primes 3*n^2-1 are odd if 2 is left out, so 3*n^2 is even, so n^2 is even, so n is even = 2*k. 3*(2*k)^2-1 = 12*k^2-1.] [From R. J. Mathar, Sep 04 2008]

A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829

p = 12; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, k]], {x, 1, 1000}]; a

A143835 a(n) = Number of x <= 10^n such that 2x^2-1 is prime.

Original entry on oeis.org

7, 45, 303, 2202, 17185, 141444, 1200975, 10448345, 92435171, 828797351, 7511268020, 68680339342

Offset: 1

Artur Jasinski, Sep 02 2008, Sep 04 2008

nonn

			a(1) = 7 because are 7 different x ={2, 3, 4, 6, 7, 8, 10} <= 10^1 where 2x^2-1 is prime = {7, 17, 31, 71, 97, 127, 199}.

Bernhard Helmes, Prime sieving on the polynomial f(n)=2n^2-1.

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143832, A143833, A143834.

l = 0; p = 2; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], l = l + 1]; If[N[Log[x]/Log[10]] == Round[N[Log[x]/Log[10]]], Print[l]; AppendTo[a, l]], {x, 1, 10000000}]; a (*Artur Jasinski*)

Added link and extended to agree with website. - Ray Chandler, Jun 30 2015

A291167 Numbers k such that psi(k) is a perfect square where psi(k) = A001615(k).

Original entry on oeis.org

1, 3, 18, 20, 22, 27, 60, 66, 70, 72, 80, 88, 92, 94, 99, 115, 119, 162, 170, 210, 212, 214, 217, 240, 243, 252, 264, 265, 276, 280, 282, 288, 308, 310, 315, 320, 322, 345, 352, 357, 368, 376, 382, 385, 423, 497, 500, 510, 517, 527, 540, 594, 596, 612, 636, 637, 642, 648, 651, 679, 680, 710, 725, 742

Offset: 1

Altug Alkan, Aug 19 2017

The product of an even number of distinct members of A066436 is in the sequence. - Robert Israel, Aug 22 2017

			60 is a term because psi(60) = 144 is a perfect square.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001615, A006532, A066436.

filter:= proc(n) issqr(n*mul(1+1/p,p=numtheory:-factorset(n))) end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 22 2017

Select[Range@ 750, IntegerQ@ Sqrt[# Sum[MoebiusMu[d]^2/d, {d, Divisors@ #}]] &] (* Michael De Vlieger, Aug 19 2017 *)

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
is(n) = issquare(a001615(n));

A076294 Consider all Pythagorean triples (X,X+7,Z); sequence gives Z values.

Original entry on oeis.org

5, 7, 13, 17, 35, 73, 97, 203, 425, 565, 1183, 2477, 3293, 6895, 14437, 19193, 40187, 84145, 111865, 234227, 490433, 651997, 1365175, 2858453, 3800117, 7956823, 16660285, 22148705, 46375763, 97103257, 129092113, 270297755, 565959257, 752403973, 1575410767

Offset: 0

Henry Bottomley, Oct 05 2002

First two terms included for consistency with A076293.

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. - Mohamed Bouhamida, Sep 09 2009

			17 is in the sequence as the hypotenuse of the (8,15,17) triangle.

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

Cf. A001653, A076293, A076295, A076296.

LinearRecurrence[{0,0,6,0,0,-1},{5,7,13,17,35,73},40] (* Harvey P. Dale, Mar 19 2019 *)

Vec((1 - x)*(5 + 12*x + 25*x^2 + 12*x^3 + 5*x^4) / (1 - 6*x^3 + x^6) + O(x^50)) \\ Colin Barker, Apr 25 2017

a(n) = 6*a(n-3)-a(n-6) = sqrt((A076293(n)^2+49)/2) = sqrt(A076295(n)^2 + A076296(n)^2).
a(3n+1) = 7*A001653(n).
G.f.: (1 - x)*(5 + 12*x + 25*x^2 + 12*x^3 + 5*x^4) / (1 - 6*x^3 + x^6). - Colin Barker, Apr 25 2017

A143831 Numbers n such that 12n^2 - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 15, 17, 20, 22, 24, 25, 26, 27, 29, 30, 35, 36, 37, 38, 39, 46, 48, 55, 59, 61, 68, 69, 72, 75, 77, 79, 82, 88, 91, 93, 94, 102, 105, 107, 108, 115, 116, 117, 118, 121, 124, 130, 134, 136, 137, 140, 149, 152, 154, 157, 158, 159, 162, 167

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830.

p = 12; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, x]], {x, 1, 1000}]; a

is(n)=isprime(12*n^2-1) \\ Charles R Greathouse IV, Feb 20 2017

A143833 Numbers n such that 14n^2 - 1 is prime.

Original entry on oeis.org

1, 4, 5, 6, 10, 11, 16, 21, 26, 34, 36, 44, 45, 49, 54, 55, 59, 65, 69, 71, 76, 80, 85, 91, 95, 96, 100, 104, 106, 110, 114, 115, 120, 121, 125, 135, 139, 166, 169, 176, 180, 190, 195, 201, 204, 206, 214, 226, 230, 231, 234, 241, 254, 256, 264, 265, 269, 270, 275, 280

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

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

Cf. A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143832.

p = 14; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, x]], {x, 1, 1000}]; a
Select[Range[300],PrimeQ[14#^2-1]&] (* Harvey P. Dale, Aug 29 2011 *)

is(n)=isprime(14*n^2-1) \\ Charles R Greathouse IV, Feb 20 2017

A157469 Positive numbers y such that y^2 is of the form x^2 + (x+97)^2 with integer x.

Original entry on oeis.org

85, 97, 113, 397, 485, 593, 2297, 2813, 3445, 13385, 16393, 20077, 78013, 95545, 117017, 454693, 556877, 682025, 2650145, 3245717, 3975133, 15446177, 18917425, 23168773, 90026917, 110258833, 135037505, 524715325, 642635573, 787056257

Offset: 1

Klaus Brockhaus, Mar 12 2009

(-13,a(1)) and (A129836(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+97)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (99+14*sqrt(2))/97 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (19491+12070*sqrt(2))/97^2 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)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

			(-13, a(1)) = (-13, 85) is a solution: (-13)^2+(-13+97)^2 = 169+7056 = 7225 = 85^2.
(A129836(1), a(2)) = (0, 97) is a solution: 0^2+(0+97)^2 = 9409 = 97^2.
(A129836(3), a(4)) = (228, 397) is a solution: 228^2+(228+97)^2 = 51984+105625 = 157609 = 397^2.

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

Cf. A129836, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157470 (decimal expansion of (99+14*sqrt(2))/97), A157471 (decimal expansion of (19491+12070*sqrt(2))/97^2).

I:=[85,97,113,397,485,593]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..50]]; // G. C. Greubel, Mar 31 2018

LinearRecurrence[{0,0,6,0,0,-1},{85,97,113,397,485,593},30] (* Harvey P. Dale, Apr 04 2013 *)

{forstep(n=-20, 800000000, [3, 1], if(issquare(2*n^2+194*n+9409, &k), print1(k, ",")))};

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=85, a(2)=97, a(3)=113, a(4)=397, a(5)=485, a(6)=593.
G.f.: (1-x)*(85 + 182*x + 295*x^2 + 182*x^3 + 85*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 97*A001653(k) for k >= 1.

A157646 Positive numbers y such that y^2 is of the form x^2 + (x+31)^2 with integer x.

Original entry on oeis.org

25, 31, 41, 109, 155, 221, 629, 899, 1285, 3665, 5239, 7489, 21361, 30535, 43649, 124501, 177971, 254405, 725645, 1037291, 1482781, 4229369, 6045775, 8642281, 24650569, 35237359, 50370905, 143674045, 205378379, 293583149, 837393701

Offset: 1

Klaus Brockhaus, Mar 11 2009

(-7,a(1)) and (A118674(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+31)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (1539+850*sqrt(2))/31^2 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)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

			(-7, a(1)) = (-7, 25) is a solution: (-7)^2+(-7+31)^2 = 49+576 = 625 = 25^2.
(A118674(1), a(2)) = (0, 31) is a solution: 0^2+(0+31)^2 = 961 = 31^2.
(A118674(3), a(4)) = (60, 109) is a solution: 60^2+(60+31)^2 = 3600+8281 = 11881 = 109^2.

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

Cf. A118674, A001653, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).

I:=[25,31,41,109,155,221]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..50]]; // G. C. Greubel, Mar 31 2018

LinearRecurrence[{0,0,6,0,0,-1},{25,31,41,109,155,221},40] (* Harvey P. Dale, Oct 12 2017 *)

{forstep(n=-8, 840000000, [1, 3], if(issquare(2*n^2+62*n+961, &k), print1(k, ",")))};

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=25, a(2)=31, a(3)=41, a(4)=109, a(5)=155, a(6)=221.
G.f.: (1-x)*(25 + 56*x + 97*x^2 + 56*x^3 + 25*x^4)/(1 - 6*x^3 + x^6).
a(3*k-1) = 31*A001653(k) for k >= 1.

cp's OEIS Frontend

A130005 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.

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A130014 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

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A143830 Primes of the form 12*n^2-1.

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A143835 a(n) = Number of x <= 10^n such that 2x^2-1 is prime.

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A291167 Numbers k such that psi(k) is a perfect square where psi(k) = A001615(k).

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A076294 Consider all Pythagorean triples (X,X+7,Z); sequence gives Z values.

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A143831 Numbers n such that 12n^2 - 1 is prime.

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A143833 Numbers n such that 14n^2 - 1 is prime.

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