cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A177859 Complement of A152412.

Original entry on oeis.org

1, 2, 5, 6, 7, 9, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 25, 27, 28, 29, 30, 31, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92
Offset: 1

Views

Author

Artur Jasinski, May 14 2010

Keywords

Crossrefs

Cf. A152412, A177760, A177761, A177763, A177764, A177765.

A177760 Values m in solutions of the Thue equation s^2 = m^5+z, sorted along increasing z.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 2, 1, 23, 2, 1, 27, 3, 1, 2, 1, 4, 2, 1, 3, 2, 7, 1, 2, 3, 1, 5, 4, 2, 1, 6, 3, 2, 1, 12, 2, 1, 3, 4, 1, 2, 5, 3, 1, 2, 4, 3, 1, 2, 1, 6, 2, 3, 21, 4, 7, 5, 8, 1, 2, 73, 3, 1, 2, 4, 3, 26, 1, 2, 5, 1, 3, 9, 10, 2, 4, 6, 20, 1, 3, 2, 11, 1, 4, 2, 5, 3, 7, 1, 2, 3, 4, 1, 6, 29, 2, 3, 5, 8, 1
Offset: 1

Views

Author

Artur Jasinski, May 13 2010

Keywords

Comments

The equation has solutions for the positive z listed in A152412.
A177761 and this sequence here show pairs (s,m) that solve given these z>0. (The case z=0 has infinitely many solutions which are not included here.)
There is no 1-to-1 relation to these z because more than one (s,m) may exist for some z, in case of which all are listed here.

Examples

			(s=59, m=5=a(57), z=356) and (s=182, m=8=a(58), z=356) are solutions associated with z = A152412(57) =356.
(s=20, m=2=a(60), z=368) and (s=45531, m=73=a(61), z=368) are solutions associated with z = A152412(59) =368.

Crossrefs

Cf. A152412, A177761.

Formula

A177761(n)^2 = a(n)^5 + A152412(k) for some k>1.

Extensions

Examples and comment on coverage of multiple solutions added - R. J. Mathar, Aug 08 2010

A177761 Values s on hyperelliptic curves s^2=m^5+z sorted along increasing z.

Original entry on oeis.org

2, 6, 3, 56, 16, 4, 7, 5, 2537, 8, 6, 3788, 17, 7, 9, 8, 33, 10, 9, 18, 11, 130, 10, 12, 19, 11, 57, 34, 13, 12, 89, 20, 14, 13, 499, 15, 14, 21, 35, 15, 16, 58, 22, 16, 17, 36, 23, 17, 18, 18, 90, 19, 24, 2021, 37, 131, 59, 182, 19, 20, 45531, 25, 20, 21, 38, 26, 3447, 21, 22
Offset: 1

Views

Author

Artur Jasinski, May 13 2010

Keywords

Crossrefs

Cf. A177760, A152412.

Formula

a(n)^2 = A177760(n)^5 + A152412(k) for some k.

A177764 a(n) = smallest possible value of m corresponding to A177763(n).

Original entry on oeis.org

1, 2, 2, 3, 55, 2, 2, 2, 2, 3, 76, 4, 3, 3, 5, 3, 4, 3, 10, 3, 7, 3, 4, 3, 3, 5, 3, 3, 3, 3, 4, 3, 3, 11, 4, 5, 377, 4, 8, 6, 4, 13, 5, 7, 4, 9, 4, 5, 14, 4, 6, 4, 42, 4, 5, 4, 7, 4, 6, 5, 8, 22, 15, 4, 4, 10, 4, 5, 4, 4, 4, 6, 4, 5, 4, 7, 4, 4, 9, 4, 4, 4
Offset: 1

Views

Author

Artur Jasinski, May 13 2010

Keywords

Crossrefs

Cf. A152412, A177760, A177761, A177763, A177764, A177765, A177766.

Extensions

Edited by N. J. A. Sloane, May 15 2010

A177765 a(n) = smallest possible value of s corresponding to A177763(n).

Original entry on oeis.org

0, 5, 4, 15, 22434, 3, 2, 1, 0, 14, 50354, 31, 13, 12, 55, 11, 30, 10, 316, 9, 129, 8, 29, 7, 6, 54, 5, 4, 3, 2, 28, 1, 0, 401, 27, 53, 2759646, 26, 180, 86, 25, 609, 52, 128, 24, 242, 23, 51, 733, 22, 85, 21, 11432, 20, 50, 19, 127, 18, 84, 49, 179, 2270, 871, 17, 16, 315
Offset: 1

Views

Author

Artur Jasinski, May 13 2010

Keywords

Crossrefs

Cf. A152412, A177760, A177761, A177763, A177764, A177765, A177766.

Extensions

Edited by N. J. A. Sloane, May 15 2010

A177763 Numbers of the form m^5-s^2, m>=1, s >= 0.

Original entry on oeis.org

1, 7, 16, 18, 19, 23, 28, 31, 32, 47, 60, 63, 74, 99, 100, 122, 124, 143, 144, 162, 166, 179, 183, 194, 207, 209, 218, 227, 234, 239, 240, 242, 243, 250, 295, 316, 341, 348, 368, 380, 399, 412, 421, 423, 448, 485, 495, 524, 535, 540, 551, 583, 608, 624, 625
Offset: 1

Views

Author

Artur Jasinski, May 13 2010

Keywords

Comments

For values m see A177764.
For values s see A177765.
For numbers that have more than one representation see A177766.

Crossrefs

Cf. A152412, A177760, A177761, A177764, A177765.

Programs

  • Mathematica
    aa = {}; bb = {}; cc = {}; Do[Do[If[((N[Sqrt[x^5 - n], 300] - Round[Sqrt[x^5 - n]])^2 < 10^-300) && (Im[Round[Sqrt[x^5 - n]]] == 0), AppendTo[aa, n]; AppendTo[bb, x]; AppendTo[cc, Round[Sqrt[x^5 - n]]]; Print[{n}, {x}, {Round[Sqrt[x^5 - n]]}]], {x, 1, 1000}], {n, 1, 1000}]; Union[aa]

Extensions

Edited by N. J. A. Sloane, May 15 2010

A177766 Terms of A177763 which have more than one such representation.

Original entry on oeis.org

7, 32, 207, 828, 1376, 1692, 2000, 2656, 3807, 5751, 5840, 7168, 7487, 7520, 7740, 7751, 18775, 19845, 20240, 21303, 22159, 23743, 25168, 29808, 30464, 30743, 32512, 32768, 34524, 38551, 45824, 46332, 47776, 49375, 53775, 54204
Offset: 1

Views

Author

Artur Jasinski, May 13 2010

Keywords

Examples

			    7 is a term:   7 = 2^5 -  5^2 and   7 =  8^5 - 181^2.
   32 is a term:  32 = 2^5 -  0^2 and  32 =  6^5 -  88^5.
  207 is a term: 207 = 3^5 -  6^2 and 207 =  6^5 -  87^2.
  828 is a term: 828 = 4^5 - 14^2 and 828 = 12^5 - 498^2.

Links

Crossrefs

Cf. A152412, A177760, A177761, A177764, A177765.

Extensions

More terms from Zak Seidov, May 14 2010
Edited by N. J. A. Sloane, May 15 2010

A152411 Nonnegative integers representable as m^2 - n^4 for positive integers m,n.

Original entry on oeis.org

0, 3, 8, 9, 15, 19, 20, 24, 33, 35, 40, 48, 51, 63, 65, 68, 73, 80, 84, 88, 99, 104, 105, 115, 120, 128, 129, 143, 144, 148, 153, 159, 163, 168, 175, 180, 185, 195, 200, 201, 208, 209, 216, 224, 225, 228, 240, 243, 255, 260, 273, 275, 280, 288, 289, 303, 304, 308, 319, 320
Offset: 1

Views

Author

N. J. A. Sloane, Oct 24 2009, based on email from Joerg Arndt, Oct 10 2009

Keywords

Comments

Nonnegative integers representable as the product u*v with (u-v)/2 being a positive square.

Links

Crossrefs

Cf. A087286, A165289, A152412.

Programs

  • Maple
    filter:= proc(x) local d,u;
  d:= select(t -> t^2 > x, numtheory:-divisors(x));
  for u in d do if issqr((u-x/u)/2) then return true fi od;
  false
end proc:
filter(0):= true:
select(filter, [$0..1000]); # Robert Israel, Nov 06 2017
  • Mathematica
    filterQ[x_] := Catch[With[{d = Select[Divisors[x], #^2 > x&]}, Do[If[IntegerQ[Sqrt[(u-x/u)/2]], Throw[True]], {u, d}]; Throw[False]]];
filterQ[0] = True;
Select[Range[0, 1000], filterQ] (* Jean-François Alcover, Jul 24 2020, after Robert Israel *)
  • PARI
    for(k=1,1000, fordiv(k,d, if(d*d>=k,break); if( issquare((k\d - d)/2), print1(k,", "); break) ) )

Extensions

Edited and extended by Max Alekseyev, Feb 06 2010

A177770 Conjectured positive numbers which have more than one representation (m,s) as a difference s^2 - m^5, m >= 1, s > 0.

Original entry on oeis.org

224, 356, 368, 433, 497, 657, 740, 1057, 1060, 1088, 1124, 1377, 1680, 1828, 2400, 2576, 3332, 3357, 3599, 4100, 4324, 4868, 7076, 8001, 8217, 10993, 11268, 11417, 12289, 13312, 14225, 14852, 15025, 15193, 15593, 15633, 17201, 19140, 20132, 20448, 21124
Offset: 1

Views

Author

Artur Jasinski, May 13 2010

Keywords

Comments

Warning: terms may be missing, see the Blomberg link.

Examples

			a(1) = 224 = 15^2-1^5 = 16^2-2^5.
a(2) = 356 = 59^2-5^5 = 182^2-8^5.
a(3) = 368 = 20^2-2^5 = 45531^2-73^5.
a(4) = 433 = 26^2-3^5 = 3447^2-26^5.
a(5) = 497 = 23^2-2^5 = 39^2-4^5.
a(6) = 657 = 30^2-3^5 = 41^2-4^5.
a(7) = 740 = 42^2-4^5 = 12842^2-44^5.
a(8) = 1057 = 332^2-2^5 = 1375^2-18^5.
a(9) = 1060 = 94^2-6^5 = 2822^2-24^5.
a(10) = 1088 = 33^2-1^5 = 184^2-8^5.
a(11) = 1124 = 34^2-2^5 = 318^2-10^5.
a(12) = 1377 = 49^2-4^5 = 1574^2-19^5.
a(13) = 1680 = 41^2-1^5 = 52^2-4^5.
a(14) = 1828 = 98^2-6^5 = 186^2-8^5 = 90298^2-96^5.

Links

  • Lars Blomberg, Table of n, a(n) for n = 1..9889. The table contains solutions with differences < 10^9 generated for m <= 10^6. No more solutions are found for m <= 10^9.

Crossrefs

Cf. A152412, A177760, A177761, A177763-A177766.

Extensions

Added constraint s>0 to the definition, inserted 368 - R. J. Mathar, May 23 2010
Name clarified, a(25)-a(41) from Lars Blomberg, Aug 13 2013
Representations for a(3) added in Example by Lars Blomberg, Aug 13 2013

A179439 Positive numbers not of the form x^5 - y^2.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 20, 21, 22, 24, 25, 26, 27, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85
Offset: 1

Views

Author

Artur Jasinski, Jul 14 2010

Keywords

Examples

			19 can be expressed as 55^5 - 22434^2, so 19 is not in the sequence.

Crossrefs

Cf. A152412, A177760, A177761, A177764, A177765, A177859.

Extensions

Edited by Arkadiusz Wesolowski, Jan 05 2013
Showing 1-10 of 12 results. Next

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