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

Author

T. D. Noe, Feb 09 2012

Keywords

• nonn
• easy

Comments

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.

Links

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

Crossrefs

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).

Programs

Mathematica

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

Formula

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