A058529 -id:A058529 - cp's OEIS frontend

A201916 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.

0, 75, 203, 323, 552, 708, 1020, 1127, 1311, 1428, 1608, 1820, 1955, 2336, 2675, 3128, 3311, 3627, 3927, 4140, 4508, 4743, 5535, 6003, 6800, 7280, 7848, 8211, 8588, 9240, 9860, 11063, 11895, 13583, 14168, 15180, 15827, 16827, 18011, 18768, 20915, 22836

Offset: 1

T. D. Noe, Feb 09 2012

Note that 2737 = 7 * 17 * 23, the product of the first three distinct primes in A058529 (and A001132) and hence the smallest such number. This sequence satisfies a linear difference equation of order 55 whose 55 initial terms can be found by running the Mathematica program.
There are many sequences like this one. What determines the order of the linear difference equation? All primes p have order 7. For those p, it appears that p^2 has order 11, p^3 order 15, and p^i order 3+4*i. It appears that for semiprimes p*q (with p > q), the order is 19. What is the next term of the sequence beginning 3, 7, 19, 55, 163? This could be sequence A052919, which is 1 + 2*3^f, where f is the number of primes.
The crossref list is thought to be complete up to Feb 14 2012.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A001652 (1), A076296 (7), A118120 (17), A118337 (23), A118674 (31).
Cf. A129288 (41), A118675 (47), A118554 (49), A118673 (71), A129289 (73).
Cf. A118676 (79), A129298 (89), A129836 (97), A157119 (103), A161478 (113).
Cf. A129837 (119), A129992 (127), A129544 (137), A161482 (151).
Cf. A206426 (161), A130608 (167), A161486 (191), A185394 (193).
Cf. A129993 (199), A198294 (217), A130609 (223), A129625 (233).
Cf. A204765 (239), A129991 (241), A207058 (263), A129626 (281).
Cf. A205644 (287), A207059 (289), A129640 (313), A205672 (329).
Cf. A129999 (337), A118611 (343), A130610 (359), A207060 (401).
Cf. A129641 (409), A207061 (433), A130645 (439), A130004 (449).
Cf. A129642 (457), A129725 (521), A101152 (569), A130005 (577).
Cf. A207075 (479), A207076 (487), A207077 (497), A207078 (511).
Cf. A111258 (601), A115135 (617), A130013 (647), A130646 (727).
Cf. A122694 (761), A123654 (809), A129010 (833), A130647 (839).
Cf. A129857 (857), A130014 (881), A129974 (937), A129975 (953).
Cf. A130017 (967), A118630 (2401), A118576 (16807).

d = 2737; terms = 100; t = Select[Range[0, 55000], IntegerQ[Sqrt[#^2 + (#+d)^2]] &]; Do[AppendTo[t, t[[-1]] + 6*t[[-27]] - 6*t[[-28]] - t[[-54]] + t[[-55]]], {terms-55}]; t

a(n) = a(n-1) + 6*a(n-27) - 6*a(n-28) - a(n-54) + a(n-55), where the 55 initial terms can be computed using the Mathematica program.

G.f.: x^2*(73*x^53 +116*x^52 +100*x^51 +171*x^50 +104*x^49 +184*x^48 +57*x^47 +92*x^46 +55*x^45 +80*x^44 +88*x^43 +53*x^42 +139*x^41 +113*x^40 +139*x^39 +53*x^38 +88*x^37 +80*x^36 +55*x^35 +92*x^34 +57*x^33 +184*x^32 +104*x^31 +171*x^30 +100*x^29 +116*x^28 +73*x^27 -363*x^26 -568*x^25 -480*x^24 -797*x^23 -468*x^22 -792*x^21 -235*x^20 -368*x^19 -213*x^18 -300*x^17 -316*x^16 -183*x^15 -453*x^14 -339*x^13 -381*x^12 -135*x^11 -212*x^10 -180*x^9 -117*x^8 -184*x^7 -107*x^6 -312*x^5 -156*x^4 -229*x^3 -120*x^2 -128*x -75) / ((x -1)*(x^54 -6*x^27 +1)). - Colin Barker, May 18 2015

A329095 Odd numbers k such that x^2 == 2 (mod k) has no solution.

Original entry on oeis.org

3, 5, 9, 11, 13, 15, 19, 21, 25, 27, 29, 33, 35, 37, 39, 43, 45, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 75, 77, 81, 83, 85, 87, 91, 93, 95, 99, 101, 105, 107, 109, 111, 115, 117, 121, 123, 125, 129, 131, 133, 135, 139, 141, 143, 145, 147, 149, 153, 155, 157, 159, 163

Offset: 1

Jianing Song, Nov 04 2019

Complement of A058529 over the odd numbers: odd numbers k such that x^2 == 2 (mod k) has solutions.
Odd numbers k such that at least one prime factor of k is congruent to 3 or 5 modulo 8 (at least one prime factor is in A003629).
Also odd terms in A025020.

			x^2 == 2 (mod 45) has no solution, so 45 is a term.

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

Cf. A003629. A047621 is a subsequence.

Cf. A058529, A057126, A025020 (numbers k such that x^2 == 2 (mod k) has no solution).

filter:= proc(t) (numtheory:-factorset(t) mod 8) intersect {3,5} <> {} end proc:
select(filter, [seq(i,i=1..1000,2)]); # Robert Israel, Nov 05 2019

Reap[Do[If[AnyTrue[FactorInteger[k][[All, 1]], MatchQ[Mod[#, 8], 3|5]&], Sow[k]], {k, 1, 999, 2}]][[2, 1]] (* Jean-François Alcover, Aug 22 2020 *)

isA329095(k) = (k%2) && !issquare(Mod(2,k))

A066507 Numbers k such that there is a solution to x^2 == 2^k (mod k).

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 10, 12, 14, 16, 17, 18, 20, 22, 23, 24, 26, 28, 30, 31, 32, 34, 36, 38, 40, 41, 42, 44, 46, 47, 48, 49, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 84, 86, 88, 89, 90, 92, 94, 96, 97, 98, 100, 102, 103, 104, 106, 108, 110

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

All even numbers are in this sequence.

Odd terms in the sequence are numbers whose prime factors are +-1 (mod 8) (A058529), i.e., odd k such that x^2 == 2 (mod k) has a solution. - Jason Earls, Jan 22 2002

Cf. A058529, A001132, A057126.

isok(n) = {for (x=0, n-1, if (Mod(x, n)^2 == Mod(2, n)^n, return (1));); return (0);} \\ Michel Marcus, Nov 20 2013

Corrected by Vladeta Jovovic, Jan 22 2002

More terms from Jason Earls, Jan 22 2002

A117472 Values of c in a^2 + b^2 = c^2, where b - a = 17 and gcd(a,b,c)=1.

Original entry on oeis.org

25, 53, 137, 305, 797, 1777, 4645, 10357, 27073, 60365, 157793, 351833, 919685, 2050633, 5360317, 11951965, 31242217, 69661157, 182092985, 406014977, 1061315693, 2366428705, 6185801173, 13792557253, 36053491345

Offset: 1

Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 19 2006

b - a = 17 is the third term in A058529 The values of 'a' are defined by A117473

			c(5) = 6*137 - 25 = 797
555*555 + 572*572 = 797*797, where 572-555 = 17 and gcd(555,572,797)=1

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

Cf. A117473, A058529.

c(1)=25, c(2)=53, c(3)=137, c(4)=305, c(n) = 6*c(n-2) - c(n-4), if c > 4

G.f.: -x*(13*x^3+13*x^2-53*x-25) / ((x^2-2*x-1)*(x^2+2*x-1)). [Colin Barker, Dec 17 2012]

A117473 The values of 'a' in a^2 + b^2 = c^2, where b - a = 17 and gcd(a, b, c) = 1.

Original entry on oeis.org

7, 28, 88, 207, 555, 1248, 3276, 7315, 19135, 42676, 111568, 248775, 650307, 1450008, 3790308, 8451307, 22091575, 49257868, 128759176, 287095935, 750463515, 1673317776, 4374021948, 9752810755, 25493668207

Offset: 1

Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 19 2006

b - a = 17 is the third term in A058529.

			a(5) = 6*88 - 7 + 34 = 555 and 555^2 + 572^2 = 797^2 and 572 - 555 = 17 and gcd(555, 572, 797) = 1.

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

Cf. A058529, A117472.

CoefficientList[Series[(5x^4 + 7x^3 - 18x^2 - 21x - 7)/((x - 1)(x^2 - 2x - 1)(x^2 + 2x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Apr 14 2017 *)
LinearRecurrence[{1,6,-6,-1,1},{7,28,88,207,555},30] (* Harvey P. Dale, Jul 31 2017 *)

a(1) = 7, a(2) = 28, a(3) = 207, a(4) = 555, a(n) = 6*a(n-2) - a(n-4) + 34.

G.f.: x*(5*x^4 + 7*x^3 - 18*x^2 - 21*x - 7) / ((x-1)*(x^2 - 2*x - 1)*(x^2 + 2*x - 1)). - Colin Barker, Dec 17 2012

A117475 The values of c in a^2 + b^2 = c^2 where b - a = 23 and gcd(a,b,c) = 1.

Original entry on oeis.org

37, 65, 205, 373, 1193, 2173, 6953, 12665, 40525, 73817, 236197, 430237, 1376657, 2507605, 8023745, 14615393, 46765813, 85184753, 272571133, 496493125, 1588660985, 2893773997, 9259394777, 16866150857, 53967707677, 98303131145

Offset: 1

Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 19 2006

b - a = 23 is the fourth term in A058529. The values of a are in A117476.

			c(5) = 6*205 - 37 = 1193 and 832^2 + 855^2 = 1193^2 and 855-832=23 and gcd(832,855,1193)=1.

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

Cf. A058529, A117476.

c(1)=37, c(2)=65, c(3)=205, c(4)=373, c(n) = 6*c(n-2) - c(n-4).

G.f.: -x*(17*x^3+17*x^2-65*x-37) / ((x^2-2*x-1)*(x^2+2*x-1)). [Colin Barker, Dec 17 2012]

A127923 Difference between squares of legs of primitive Pythagorean triangles, sorted (with multiplicity).

Original entry on oeis.org

7, 41, 119, 161, 239, 527, 721, 959, 1081, 1241, 1393, 1519, 2047, 3281, 3479, 3713, 4207, 4633, 4681, 4879, 5593, 6647, 6887, 7327, 8119, 9401, 9641, 10199, 11753, 12121, 12319, 12593, 16999, 19159, 19199, 19873, 20447, 22393, 23359, 24521, 24521

Offset: 1

Lekraj Beedassy, Feb 06 2007

nonn

This is the sorted sequence of all products A120681(i)*A120682(i). - R. J. Mathar, Sep 24 2007
The sequence is conjectural (and may miss entries) because it is generated from a finite list of primitive Pythagorean triangles. The associated lengths in a^2+b^2=c^2 are (a,b)=(3,4), (21,20), (5,12), (15,8), (119,120), (7,24), (55,48), (65,72), (35,12), (45,28), (697,696), (9,40), (33,56), (105,88), (11,60), (63,16), (297,304), (77,36), (91,60), (39,80), (403,396), (133,156), (13,84), (207,224), (4059, 4060), (99,20), (171,140), (85,132), (117,44), (275,252), (15,112), (153,104), (51,140), (555,572), (95,168), (143,24), (17,144), (253, 204), (225,272), (165,52), (1755,1748), (429,460),... with gcd(a, b)=1 and |a^2-b^2| in the sequence. - R. J. Mathar, Sep 24 2007
Confirmed sequence is accurate and complete. Observe that both b-a and b+a must be in A058529. Running through the possible combinations of those values with products below 25000 that produce values of a and b that are legs of primitive Pythagorean triangles confirms list is correct. Note that terms of this sequence must also be in A058529. - Ray Chandler, Apr 11 2010
24521 appears twice in the sequence for (a,b)=(52,165) and (1748,1755). - Ray Chandler, Apr 11 2010

Cf. A058529, A120681, A120682.

More terms from R. J. Mathar, Sep 24 2007

Removed "conjectural" from description by Ray Chandler, Apr 11 2010

A152910 Difference between legs of primitive Pythagorean triangles sorted first on hypotenuse, then short leg.

Original entry on oeis.org

1, 7, 7, 17, 1, 23, 31, 17, 49, 47, 23, 7, 71, 41, 41, 7, 79, 31, 97, 73, 17, 127, 119, 89, 47, 1, 113, 161, 119, 49, 73, 167, 103, 23, 199, 31, 161, 103, 89, 223, 241, 151, 191, 137, 71, 79, 217, 137, 17, 287, 233, 287, 49, 113, 119, 47, 337, 281, 23, 217, 193, 151, 97

Offset: 1

Paul Curtz, Dec 15 2008, Dec 19 2008

nonn

M. de Frénicle, Méthode pour trouver la solution des problèmes par les exclusions, in: Divers ouvrages de mathématique et de physique par messieurs de l'académie royale des sciences, (1693) pp 1-44, table p 31.

Cf. A058529 (gives range), A120682 (different order).

Minor edits by Omar E. Pol, Jan 06 2009

Edited and extended by Ray Chandler, Apr 10 2010

A201917 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+84847)^2 = y^2.

Original entry on oeis.org

0, 2085, 2325, 5253, 6141, 6293, 7728, 10013, 11960, 12920, 14637, 16940, 17112, 18737, 21648, 21948, 23541, 24633, 26588, 27716, 31620, 33012, 34937, 35145, 38012, 40641, 42716, 44268, 47633, 49848, 52785, 54237, 56420, 56840, 60605, 63828, 67541, 70448

Offset: 1

T. D. Noe, Feb 09 2012

nonn

Note that 84847 = 7 * 17 * 23 * 31, the first four primes in A058529.

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 159 terms from Vincenzo Librandi)

Cf. A201916 (has list of all such sequences).

d = 84847; t = Select[Range[0,68000], IntegerQ[Sqrt[#^2 + (#+d)^2]] &]

a(n) = a(n-1) + 6*a(n-81) - 6*a(n-82) - a(n-162) + a(n-163), where the 163 initial terms can be computed using the Mathematica program. The initial terms begin with 0 and end with 1696940.

A306236 a(n) is the smallest integer m > n with integer j > m makes n^2, m^2 and j^2 an arithmetic progression.

Original entry on oeis.org

5, 10, 15, 20, 25, 30, 13, 40, 45, 50, 55, 60, 65, 26, 75, 80, 25, 90, 95, 100, 39, 110, 37, 120, 125, 130, 135, 52, 145, 150, 41, 160, 165, 50, 65, 180, 185, 190, 195, 200, 85, 78, 215, 220, 225, 74, 65, 240, 61, 250, 75, 260, 265, 270, 275, 104, 285, 290

Offset: 1

Jinyuan Wang, Feb 08 2019

nonn

a(n) and n have the same parity.
If k is a term in A058529, gcd(k, a(k)) does not necessarily equal 1. For example, k = 217, 289, 343, 497, 529, 553, 679, 889, 961, 1127, ...
Conjecture: if gcd(k, a(k)) = 1, then k is a term in A058529.
Proof: if k is not in A058529, then k either is even or has a prime factor p == 3, 5 (mod 8). If k is even, then a(k) is also even, so 2 divides gcd(k, a(k)). If k has a prime factor p == 3, 5 (mod 8), then 2*m^2 == j^2 (mod p), 2^((p-1)/2)*m^(p-1) == -m^(p-1) == j^(p-1) (mod p), so m and j must both be multiples of p. As a result, p divides gcd(k, a(k)). - Jianing Song, Feb 09 2019

			a(1) = 5 because 1^2, 5^2 and 7^2 are an arithmetic progression.

Cf. A003629, A058529, A289398 (integer j).

Array[Block[{m = # + 2}, While[! IntegerQ@ Sqrt[2 m^2 - #^2], m += 2]; m] &, 58] (* Michael De Vlieger, Feb 15 2019 *)

a(n) = {m=n+2; while(issquare(2*m^2-n^2)==0, m=m+2); m;}

a(n) = sqrt((n^2 + A289398(n)^2)/2).
For positive integer k, a(2*k^2 - 1) = 2*k^2 + 2*k + 1.
a(A003629(k)) = 5*A003629(k).
a(n) <= 5*n.
a(k*n) = k*a(n) for all k not in A058529. - Jianing Song, Feb 15 2019

cp's OEIS Frontend

A201916 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.

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A329095 Odd numbers k such that x^2 == 2 (mod k) has no solution.

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A066507 Numbers k such that there is a solution to x^2 == 2^k (mod k).

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A117472 Values of c in a^2 + b^2 = c^2, where b - a = 17 and gcd(a,b,c)=1.

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A117473 The values of 'a' in a^2 + b^2 = c^2, where b - a = 17 and gcd(a, b, c) = 1.

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A117475 The values of c in a^2 + b^2 = c^2 where b - a = 23 and gcd(a,b,c) = 1.

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A127923 Difference between squares of legs of primitive Pythagorean triangles, sorted (with multiplicity).

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A152910 Difference between legs of primitive Pythagorean triangles sorted first on hypotenuse, then short leg.

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A201917 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+84847)^2 = y^2.

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