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 10 results.

A050800 Values of n^2 + 1 resulting from A050796.

Original entry on oeis.org

2, 50, 65, 145, 170, 290, 325, 442, 485, 530, 730, 785, 901, 962, 1025, 1090, 1157, 1370, 1445, 1682, 1765, 1850, 1937, 2117, 2210, 2305, 2501, 2705, 2810, 3026, 3250, 3365, 3601, 3845, 3970, 4097, 4490, 4625, 4901, 5185, 5330, 5626, 5777, 5930, 6085
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Crossrefs

Cf. A050796.

Programs

  • Mathematica
    t={2}; Do[i=c=2; x=n^2+1; While[iJayanta Basu, Jun 01 2013 *)

A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

Original entry on oeis.org

3, 9, 17, 19, 33, 35, 51, 73, 81, 99, 105, 129, 145, 147, 161, 163, 179, 195, 201, 233, 243, 273, 289, 291, 297, 339, 361, 387, 393, 451, 465, 467, 483, 489, 513, 521, 577, 579, 585, 611, 627, 649, 675, 721, 723, 739, 777, 801, 809, 819, 849, 883, 899, 915
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Comments

Analogous solutions exist for the sum of two identical squares z^2-1 = 2.r^2 (e.g. 99^2-1 = 2.70^2). Values of 'z' are the terms in sequence A001541, values of 'r' are the terms in sequence A001542.
Looking at a^2 + b^2 = c^2 - 1 modulo 4, we must have a and b even and c odd. Taking a = 2u, b = 2v and c = 2w - 1 and simplifying, we get u^2 + v^2 = w(w+1). - Franklin T. Adams-Watters, May 19 2008
If n is in this sequence, then so is n^(2^k), for all k >= 0. - Altug Alkan, Apr 13 2016

Examples

			E.g. 51^2 - 1 = 10^2 + 50^2 = 22^2 + 46^2 = 34^2 + 38^2.

Links

Crossrefs

Cf. A050796, A050797, A001541, A001542, A001333.
Cf. A140612, A002378.

Programs

  • Mathematica
    t={}; Do[i=c=1; While[iJayanta Basu, Jun 01 2013 *)
Select[Range@ 1000, Length[PowersRepresentations[#^2 - 1, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 13 2016 *)
  • PARI
    select( {is_A050795(n)=#qfbsolve(Qfb(1,0,1),n^2-1,2)}, [1..999]) \\ M. F. Hasler, Mar 07 2022
  • Python
    from itertools import islice, count
from sympy import factorint
def A050795_gen(startvalue=2): # generator of terms >= startvalue
    for k in count(max(startvalue,2)):
        if all(map(lambda d: d[0] % 4 != 3 or d[1] % 2 == 0, factorint(k**2-1).items())):
            yield k
A050795_list = list(islice(A050795_gen(),20)) # Chai Wah Wu, Mar 07 2022

Formula

a(n) = 2*A140612(n) + 1. - Franklin T. Adams-Watters, May 19 2008
{k : A025426(k^2-1)>0}. - R. J. Mathar, Mar 07 2022

A299707 Numbers m such that m^2 + 1 can be expressed in more than one way as j^2 + k^2 with j > k > 1.

Original entry on oeis.org

18, 32, 38, 43, 47, 57, 68, 70, 72, 73, 82, 83, 93, 98, 99, 107, 112, 117, 118, 122, 123, 128, 132, 133, 138, 142, 143, 148, 157, 162, 168, 172, 173, 174, 177, 182, 183, 187, 191, 192, 193, 200, 203, 207, 208, 212, 213, 216, 217, 218, 228, 232, 233, 237, 242, 243, 251, 252
Offset: 1

Views

Author

Hugo Pfoertner, Feb 27 2018

Keywords

Examples

			a(1) = 18: 18^2 + 1 = 325 = 17^2 + 6^2 = 15^2 + 10^2,
a(2) = 32: 32^2 + 1 = 1025 = 31^2 + 8^2 = 25^2 + 20^2,
a(5) = 47: 47^2 + 1 = 2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2.

Links

Crossrefs

Cf. A050796, A299708, A300161, A300162, A300163, A300164, A300165, A300166.

A299708 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1.

Original entry on oeis.org

325, 1025, 1445, 1850, 2210, 3250, 4625, 4901, 5185, 5330, 6725, 6890, 8650, 9605, 9802, 11450, 12545, 13690, 13925, 14885, 15130, 16385, 17425, 17690, 19045, 20165, 20450, 21905, 24650, 26245, 28225, 29585, 29930, 30277, 31330, 33125, 33490
Offset: 1

Views

Author

Hugo Pfoertner, Feb 27 2018

Keywords

Examples

			a(1) = 325 = A299707(1)^2 + 1 = 18^2 + 1 is expressible in two ways:
  325 = 17^2 + 6^2 = 15^2 + 10^2.

Links

Crossrefs

Cf. A050796, A299707, A300161, A300162, A300163, A300164, A300165, A300166.

A300161 Numbers n such that n^2 + 1 can be expressed as j^2 + k^2, j > k > 1, in more ways than for any smaller n.

Original entry on oeis.org

8, 18, 47, 132, 242, 268, 993, 1568, 3957, 5257, 14318, 18543, 43932, 66347, 72662, 161832, 330182, 1413443, 1732593, 2298668, 3315268, 9548768, 15926318, 24310918, 27028568, 51853693, 162166243, 420024818, 472936732, 599832943, 1892369318
Offset: 1

Views

Author

Hugo Pfoertner, Feb 27 2018

Keywords

Examples

			a(1) = 8: 8^2 + 1 = 65 = 7^2 + 4^2,
a(2) = 18: 18^2 + 1 = 325 = 17^2 + 6^2 = 15^2 + 10^2,
a(3) = 47: 47^2 + 1 = 2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2,
a(4) = 132: 132^2 + 1 = 17425 = 129^2 + 28^2 = 127^2 + 36^2 = 120^2 + 55^2 = 116^2 + 63^2 = 105^2 + 80^2.

Crossrefs

Cf. A050796, A299707, A299708, A300162, A300163.

Extensions

a(17) from Hugo Pfoertner, Mar 08 2018
a(18)-a(21) from Robert Price, Mar 10 2018
a(22)-a(31) from Giovanni Resta, Mar 13 2018

A300165 Numbers m such that m^2 + 1 can be expressed in more than one way as j^2 + k^2 with j > k > 1 and gcd(j,k) = 1.

Original entry on oeis.org

47, 72, 73, 83, 98, 112, 122, 123, 128, 132, 133, 138, 142, 148, 157, 162, 172, 173, 174, 177, 183, 187, 191, 192, 200, 203, 208, 212, 213, 216, 217, 228, 233, 237, 242, 252, 253, 255, 265, 268, 273, 278, 288, 293, 294, 302, 307, 313, 317, 319
Offset: 1

Views

Author

Hugo Pfoertner, Feb 27 2018

Keywords

Comments

The sequence differs from A299707 by the gcd condition, which excludes representations like 18^2 + 1 = 15^2 + 10^2, 32^2 + 1 = 25^2 + 20^2, 38^2 + 1 = 34^2 + 17^2.

Examples

			a(1) = 47 because its 3 representations satisfy the conditions j > k > 1 and gcd(j,k) = 1: 47^2 + 1 = 2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2.

Links

Crossrefs

Cf. A050796, A299707, A300166, A300167, A300168.

A300166 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1 and gcd(j,k) = 1.

Original entry on oeis.org

2210, 5185, 5330, 6890, 9605, 12545, 14885, 15130, 16385, 17425, 17690, 19045, 20165, 21905, 24650, 26245, 29585, 29930, 30277, 31330, 33490, 34970, 36482, 36865, 40001, 41210, 43265, 44945, 45370, 46657, 47090, 51985, 54290, 56170, 58565, 63505
Offset: 1

Views

Author

Hugo Pfoertner, Feb 27 2018

Keywords

Comments

The sequence differs from A299708 by the gcd condition, which excludes representations like 325 = 18^2 + 1^2 = 15^2 + 10^2, 1025 = 32^2 + 1 = 25^2 + 20^2, 1445 = 38^2 + 1 = 34^2 + 17^2.

Examples

			a(1) = 2210 because its 3 representations satisfy the conditions j > k > 1 and gcd(j,k) = 1: 2210 = 47^2 + 1 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2.

Links

Crossrefs

Cf. A050796, A299707, A299708, A300165, A300167, A300168.

A050798 Numbers n such that m = n^2 + 1 is expressible as the sum of two nonzero squares in exactly two ways.

Original entry on oeis.org

1, 7, 8, 12, 13, 17, 21, 22, 23, 27, 28, 30, 31, 33, 34, 37, 41, 42, 44, 46, 48, 50, 52, 53, 55, 58, 60, 62, 63, 64, 67, 75, 76, 77, 78, 80, 81, 86, 87, 88, 89, 91, 92, 96, 97, 100, 102, 103, 104, 105, 106, 108, 109, 111, 113, 114, 115, 119, 125, 127, 129, 135, 136
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Comments

Of course m = n^2 + 1 is the sum of two squares, by definition. Here there should be just one other way to write m as a different sum of two squares.
Let p and q be primes of the form 1+4k. Then n^2+1 must be pq or 2pq. - T. D. Noe, May 27 2008

Examples

			E.g., 111^2 + 1 = 21^2 + 109^2 only.

Links

Crossrefs

Cf. A000161, A050797, A050796.

Programs

  • Mathematica
    ok[1] = True; ok[n_] := Length[ {ToRules[ Reduce[ 1 < x <= y && n^2 + 1 == x^2 + y^2, {x, y}, Integers] ] } ] == 1; Select[ Range[136], ok] (* Jean-François Alcover, Feb 16 2012 *)

Extensions

Better definition from T. D. Noe, May 27 2008

A300164 Numbers of the form n^2+1 not expressible as j^2+k^2 with j>k>1.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 50, 82, 101, 122, 197, 226, 257, 362, 401, 577, 626, 677, 842, 1226, 1297, 1522, 1601, 1682, 2026, 2402, 2602, 2917, 3137, 3482, 3722, 4226, 4357, 4762, 5042, 5477, 6242, 7057, 7226, 8101, 8837, 9026, 10202, 12101, 13457, 14401
Offset: 1

Views

Author

Hugo Pfoertner, Feb 27 2018

Keywords

Examples

			The first numbers of the form n^2 + 1 not in the sequence are:
  65 = 8^2 + 1 because it can be expressed as 65 = 7^2 + 4^2,
  145 = 12^2 + 1 = 9^2 + 8^2,
  170 = 13^2 + 1 = 11^2 + 7^2.

Links

Crossrefs

Cf. A050796, A065876, A071557, A299708.

A096079 Near hypotenuse numbers, i.e., n such that n^2 -+ 1 = a^2 + b^2, with a, b > 1.

Original entry on oeis.org

3, 7, 8, 9, 12, 13, 17, 18, 19, 21, 22, 23, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 60, 62, 63, 64, 67, 68, 70, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 86, 87, 88, 89, 91, 92, 93, 96, 97, 98, 99, 100, 102, 103, 104, 105
Offset: 1

Views

Author

Lekraj Beedassy, Jul 21 2004

Keywords

Comments

When a=b, entries are those of A001333. - Lekraj Beedassy, Aug 06 2004

Examples

			8 and 9 are in the sequence because we have 8^2 + 1 = 4^2 + 7^2; 9^2 - 1 = 4^2 + 8^2.

Crossrefs

Cf. A001481, A009003.
Union of A050795 and A050796. - Franklin T. Adams-Watters, Apr 26 2009

Extensions

Extended by Ray Chandler, Jul 29 2004
Showing 1-10 of 10 results.

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