A230711
Values of y such that x^2 + y^2 = 5^n with x and y coprime and 0 < x < y.
Original entry on oeis.org
2, 4, 11, 24, 41, 117, 278, 527, 1199, 3116, 6469, 11753, 33802, 76443, 136762, 354144, 873121, 1721764, 3565918, 9653287, 20783558, 34867797, 103232189, 242017776, 451910159, 1064447283, 2726446322, 5583548873, 10513816601, 29729597084, 66349305331
Offset: 1
a(4)=24 because 7^2+24^2=625=5^4.
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Table[Select[PowersRepresentations[5^n, 2, 2], CoprimeQ[#[[1]], #[[2]]] &][[1,2]], {n, 33}] (* T. D. Noe, Nov 04 2013 *)
A230712
Values of x such that x^2 + y^2 = 37^n with x and y coprime and 0 < x < y.
Original entry on oeis.org
1, 12, 107, 840, 5646, 27755, 124158, 462961, 961686, 5589325, 102654282, 1025046359, 8502347874, 64101459205, 356029844147, 1681548425760, 7005476875681, 21848430755052, 2978524660427, 772649642011800
Offset: 1
a(3)=107 because 107^2+198^2=50653=37^3.
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Table[Select[PowersRepresentations[37^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)
A230743
Values of x such that x^2 + y^2 = 41^n with x and y coprime and 0 < x < y.
Original entry on oeis.org
4, 9, 115, 720, 2476, 42471, 4765, 1788961, 3780956, 51872200, 310486445, 1142532559, 18483128564, 4205436520, 799862636324, 1584162310079, 23384002313285, 133802323596440, 526151093402156, 8041209044472401, 2783579583540395, 357525366658772391
Offset: 1
a(3)=115 because 115^2+236^2=68921=41^3.
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Table[Select[PowersRepresentations[41^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)
A230759
Values of x such that x^2 + y^2 = 53^n with x and y coprime and 0 < x < y.
Original entry on oeis.org
2, 28, 259, 1241, 3647, 14715, 399301, 4810319, 34161842, 146769868, 244200526, 4359995640, 73982566838, 804676166812, 4381447604821, 15981352647839, 8477785985767, 965700694136205, 13070487060661219, 114948480102611400, 541029996598203398
Offset: 1
a(3)=259 because 259^2+286^2=148877=53^3.
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Table[Select[PowersRepresentations[53^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)
A230841
Values of x such that x^2 + y^2 = 61^n with x and y coprime and 0 < x < y.
Original entry on oeis.org
5, 11, 234, 1320, 9475, 117469, 266286, 9184560, 3302155, 520632300, 1387108806, 23922442439, 154165737965, 933420304380, 13338456688674, 22995028210081, 1026964091673115, 713853567388260, 54078566783400895, 171226928056302601, 2435077776719657394
Offset: 1
a(3)=234 because 234^2+415^2=226981=61^3.
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Table[Select[PowersRepresentations[61^n, 2, 2], CoprimeQ[#[[1]], #[[2]]] &][[1, 1]], {n, 22}] (* T. D. Noe, Nov 04 2013 *)
Table[Select[PowersRepresentations[61^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)
a(10) and a(20) corrected by
Zak Seidov, Nov 02 2013
A230962
Values of x such that x^2 + y^2 = 73^n with x and y coprime and 0 < x < y.
Original entry on oeis.org
3, 48, 296, 721, 10072, 213785, 1958709, 7613760, 21165597, 894454032, 12278087704, 59926173839, 62518379032, 3374316625735, 58552907681096, 416603004343680, 1261259807092797, 10231862403603888, 255781764375436389, 2697529798981443601, 11543491568219853608
Offset: 1
a(3) = 296 because 296^2 + 549^2 = 389017 = 73^3.
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f:=n -> min([abs@Re,abs@Im]((3+8*I)^n)):
map(f, [$1..50]); # Robert Israel, Mar 31 2017
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Table[Select[PowersRepresentations[73^n, 2, 2], CoprimeQ@@#&][[1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Mar 02 2014 *)
Table[Min[Abs[Re[(3 + 8I)^n]], Abs[Im[(3 + 8I)^n]]], {n, 30}] (* Indranil Ghosh, Mar 31 2017, after formula by Robert Israel *)
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from sympy import I, re, im
print([min(abs(re((3 + 8*I)**n)), abs(im((3 + 8*I)**n))) for n in range(1, 31)]) # Indranil Ghosh, Mar 31 2017, after formula by Robert Israel
A341678
Irregular triangle read by rows: row n consists of all numbers x such that x^2 + y^2 = A006278(n), with 0 < x < y.
Original entry on oeis.org
1, 1, 4, 4, 9, 12, 23, 2, 19, 46, 67, 74, 86, 109, 122, 64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743, 59, 132, 531, 581, 627, 876, 1008, 1284, 1588, 1659, 1723, 2092, 2136, 2317, 2373, 2736, 2757, 2803, 3072, 3164, 3333, 3469, 3704, 3821, 4028, 4077, 4136, 4371, 4596, 4668, 4712, 4851
Offset: 1
Triangle starts:
1,
1, 4,
4, 9, 12, 23,
2, 19, 46, 67, 74, 86, 109, 122,
64, 103, 167, 191, 236, 281, 292, 359, 449, 512, 568, 601, 607, 664, 673, 743,
...
In the second row, calculations are as follows. 5*13 is the product of the first two primes congruent to 1 (mod 4), and 65 = 1^2 + 8^2 = 4^2 + 7^2, so the second row is 1, 4.
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row(n) = {my(t=1, q=3, v=vector(2^n/2)); for(k=1, n, until(q%4==1, q=nextprime(q+1)); t*=q); q=0; for(k=1, #v, until(issquare(t-q^2), q++); v[k]=q); v; } \\ Jinyuan Wang, Mar 03 2021
A348635
a(n) is the smallest positive number k coprime to (2n+1)!! such that (2n+1)!! + k^2 is a square.
Original entry on oeis.org
1, 1, 4, 4, 29, 17, 436, 356, 569, 1847, 27704, 72944, 1283333, 726079, 23833532, 45232276, 302068799, 616565857, 26369361188, 23157514888, 70991664061, 505527042479, 1150735735948, 13238389944712, 58668785675111, 209280259070287, 7809609503808088, 530566746979816
Offset: 1
a(5)=29 since 106^2 - 29^2 = 10395 = 3*5*7*9*11 and 29 is relatively prime to 10395 and is as small as possible.
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df(n) = (2*n)! / n! / 2^n; \\ A001147
a(n) = my(d=df(n+1), k=1); while (!((gcd(d,k)==1) && issquare(d+k^2)), k++); k; \\ Michel Marcus, Jan 06 2022
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df(n) = (2*n)! / n! / 2^n; \\ A001147
a(n) = my(d=df(n+1), m=sqrtint(d), k); while (!(issquare(m^2-d, &k) && gcd(d,k)==1), m++); k; \\ Michel Marcus, Jan 06 2022
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