A184763 -id:A184763 - cp's OEIS frontend

A184886 Irregular triangle of the square root of the sums of squares mentioned in A184763.

70, 5, 8075, 17267, 92, 143, 245, 1518, 60207, 106, 236818619, 2001863, 652, 679, 3406, 138, 225643, 77, 29, 158, 3128, 778998480, 724, 413, 331668, 182, 195, 357, 8033, 9010, 126035, 1835940, 253, 4017, 385, 788, 1612, 8687, 150878, 8758575, 33158210, 143, 531, 3770, 10384, 751228, 274, 495, 1281, 1133000, 1012, 4433, 121268, 56855, 440, 106403, 3069, 2725, 16332, 655, 453765, 1525, 1997277, 4035066, 430, 2619, 2420957, 795, 3465

Offset: 1

T. D. Noe, Jan 24 2011

Sequence A184762 gives the length of row n. A180442 lists the nonempty rows. These numbers are in Table 3 of the paper by Bremner, Stroeker, and Tzanakis.

			The triangle is
(row 1)  70
(row 3)  5, 8075, 17267
(row 7)  92, 143, 245, 1518, 60207
(row 9)  106, 236818619
(row 11) 2001863
(row 13) 652
(row 15) 679, 3406

A. Bremner, R. J. Stroeker, N. Tzanakis, On Sums of Consecutive Squares, J. Number Theory 62 (1997), 39-70.

A180442 Numbers n such that a sum of two or more consecutive squares beginning with n^2 is a square.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 15, 17, 18, 20, 21, 22, 25, 27, 28, 30, 32, 38, 44, 50, 52, 55, 58, 60, 64, 65, 67, 73, 74, 76, 83, 87, 91, 103, 104, 106, 112, 115, 117, 119, 121, 124, 128, 129, 131, 132, 137, 140, 142, 146, 158, 168, 170, 172, 175, 178, 181, 183, 192, 193, 197, 199, 200, 204

Offset: 1

Zhining Yang, Jan 19 2011

That is, numbers n such that Sum_{i=n..k} i^2 is a square for some k > n.
The paper by Bremner, Stroeker, and Tzanakis describes how they found all n <= 100 by solving elliptic curves. Their solutions are the same as the terms in this sequence. They also show that there are only a finite number of sums of squares beginning with n^2 that sum to a square. For example, starting with 3^2, there are only 3 ways to sum consecutive squares to produce a square: 3^2 + 4^2, 3^2 + ... + 580^2, and 3^2 + ... + 963^2. See A184762, A184763, A184885, and A184886 for more results from their paper.
This sequence is more difficult than A001032, which has the possible lengths of the sequences of consecutive squares that sum to a square. Be careful adding terms to this sequence; a simple search may miss some terms. An elliptic curve needs to be solved for each number.
It is conjectured that the sequence continues 103, 104, 106, 112, 115, 117, 119, 121, 124, 128, 129, 131, 132, 137, 140, 142, 146, 158, 168, 170, 172, 175, 178, 181, 183, 192, 193, 197, 199, 200. - Jean-François Alcover, Sep 17 2013. Conjecture confirmed (see the Schoenfield link below). - Jon E. Schoenfield, Nov 22 2013

			30 is in the sequence because 30^2 + 31^2 + 32^2 + ... + 197^2 + 198^2 = 1612^2.

Jon E. Schoenfield, Table of n, a(n) for n = 1..123 (includes a derivation of the elliptic curves and Magma code used to find the terms)
A. Bremner, R. J. Stroeker, N. Tzanakis, On Sums of Consecutive Squares, J. Number Theory 62 (1997), 39-70.
K. S. Brown, Sum of Consecutive Nth Powers Equals an Nth Power
Masoto Kuwata, Jaap Top, An elliptic surface related to sums of consecutive squares, Exposition. Math. 12 (1994) 181-192

Cf. A180259, A180273, A180274, A180465.

Cf. A075404, A075405, A075406.

jmax[11] = jmax[74] = 10^5; jmax[n_ /; n > 91] = 10^6; jmax[] = 10^4; Reap[For[n = 1, n <= 200, n++, s = n^2; For[j = n+1, j <= jmax[n], j++, s += j^2; If[IntegerQ[Sqrt[s]], Sow[n]; Print[n, "(", j, ", ", Sqrt[s], ")"]; Break[]]]]][[2, 1]] (* _Jean-François Alcover, Sep 17 2013, translated and adapted from Pari *)

for(n=1, 100,s=n^2;for(j=n+1,999999,s+=j^2; if(issquare(s), print1(n, "(",j,",",sqrtint(s),")");break())))

Numbers n such that A075404(n) > 0.

Example simplified by Jon E. Schoenfield, Sep 18 2013

More terms from Jon E. Schoenfield, Nov 22 2013

A075404 Smallest m > n such that Sum_{i=n..m} i^2 is a square, or 0 if no such m exists.

Original entry on oeis.org

24, 0, 4, 0, 0, 0, 29, 0, 32, 0, 22908, 0, 108, 0, 111, 0, 39, 28, 0, 21, 116, 80, 0, 0, 48, 0, 59, 77, 0, 198, 0, 609, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 67, 0, 0, 0, 0, 0, 171, 0, 147, 0, 0, 3533, 0, 0, 2132, 0, 92, 0, 0, 0, 305, 282, 0, 116, 0, 0, 0, 0, 0, 194, 36554, 0, 99, 0, 0, 0, 0, 0, 0, 276, 0, 0, 0, 136, 0, 0, 0, 332, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Zak Seidov, Sep 13 2002

nonn

For a(1) see A000330.
The corresponding squares are in A075405, the numbers of terms in the sum = a(n)-n+1 are in A075406.
All terms were verified by solving elliptic curves. If a(n)>0, then there may be additional values of m that produce squares. See A184763 for more information.

			a(1) = 24 because 1^2+...+24^2 = 70^2, a(7) = 29 because 7^2+...+29^2 = 92^2.

See A180442.

Cf. A000330, A075405, A075406, A180442 (n such that a(n) > 0).

s[n_,k_]:=Module[{m=n+k-1},(m(m+1)(2m+1)-n(n-1)(2n-1))/6]; mx=40000; Table[k=2; While[k

Corrected and extended by Lior Manor, Sep 19 2002

Corrected and edited by T. D. Noe, Jan 21 2011

A184762 The number of numbers k > n such that Sum_{i=n..k} i^2 is a square.

Original entry on oeis.org

1, 0, 3, 0, 0, 0, 5, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 2, 0, 0, 7, 0, 2, 2, 0, 1, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

T. D. Noe, Jan 21 2011

It is an old result (see Watson) that for n=1 the only k>n is k=24. Bremner, Stroeker, and Tzanakis compute the k for n <= 100 by solving elliptic curves. This sequence lists the number of k for each n; the values of k are in A184763. Sequence A180442 lists the n for which a(n) is nonzero.

A. Bremner, R. J. Stroeker, N. Tzanakis, On Sums of Consecutive Squares, J. Number Theory 62 (1997), 39-70.
G. N. Watson, The problem of the square pyramid, Messenger of Mathematics 48 (1918), pp. 1-22.

Cf. A075404, A075405, A075406.

A180259 Squares which are the sum of consecutive squares starting with 25^2.

Original entry on oeis.org

625, 33124, 38025, 127449, 64529089, 81180100, 15884821225, 3370675683600

Offset: 1

Zhining Yang, Jan 17 2011

That is, terms are squares of the form sum_{i=25..m} i^2 = (m-24) *(2*m^2+51*m+1225) / 6 for some m. Known solutions refer to m = 25, 48, 50, 73, 578, 624, 3625 and 21624, and no further in the range m <= 70000000.

This sequence is complete. See A180442 and A184763.

			38025 is in the sequence because 38025 = 195^2 = 25^2 + 26^2 + ... + 50^2.

Cf. A001032, A094196, A151557, A180273.

Select[Accumulate[Range[25, 22000]^2], IntegerQ[Sqrt[#]] &] (* Harvey P. Dale, Aug 10 2023 *)

for(n=26,9999999,t=n*(n+1)*(2*n+1)/6-4900;if(issquare(t),print1(t,",")))

A180273 Squares which are a sum of consecutive squares starting with 7^2.

Original entry on oeis.org

49, 8464, 20449, 60025, 2304324, 3624882849

Offset: 1

Zhining Yang, Jan 17 2011

That is, terms are squares of the form sum_{i=7..m} i^2 for some m. This sequence is complete. See A180442 and A184763.

			a(2) = 8464 = 92^2 = 7^2+8^2+9^2+...+28^2+29^2.

Cf. A001032, A094196, A151557, A180259.

Select[Accumulate[Range[7,3000]^2],IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Jul 16 2014 *)

for(n=8,9999999,t=n*(n+1)*(2*n+1)/6-91;if(issquare(t),print1(t,",")))

A184885 Irregular triangle in which row n has the values of k>1 such that sum_{i=n..n+k-1} i^2 is a square.

Original entry on oeis.org

24, 2, 578, 961, 23, 33, 50, 184, 2209, 24, 552049, 22898, 96, 97, 312, 23, 5329, 11, 2, 24, 289, 1221025, 96, 59, 6889, 24, 26, 49, 554, 600, 3601, 21600, 33, 338, 50, 96, 169, 578, 4056, 61250, 148825, 11, 59, 312, 649, 11881, 24, 50, 122, 15625, 96, 338, 3479, 2075, 33, 3179, 242, 218, 864, 50, 8450, 122, 22801, 36481, 24, 194, 25921, 50, 242

Offset: 1

T. D. Noe, Jan 24 2011

That is, this sequence gives the number of squares in the sums described in A184763. Sequence A184762 gives the length of row n. A180442 lists the nonempty rows. All these numbers must appear in A001032.

			The triangle is
(row 1)  24
(row 3)  2, 578, 961
(row 7)  23, 33, 50, 184, 2209
(row 9)  24, 552049
(row 11) 22898
(row 13) 96
(row 15) 97, 312

Row n = row n of A184763 minus n-1.

A180465 Squares which are a sum of consecutive squares starting with 38^2.

Original entry on oeis.org

1444, 20449, 281961, 14212900, 107827456, 564343507984

Offset: 1

Zhining Yang, Jan 19 2011

This sequence is complete. See A180442 and A184763.

Cf. A001032, A094196, A151557, A180259, A180273

for(n=38, 9999999, t=n*(n+1)*(2*n+1)/6-17575; if(issquare(t), print1(t, ", ")))

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A184886 Irregular triangle of the square root of the sums of squares mentioned in A184763.

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A180442 Numbers n such that a sum of two or more consecutive squares beginning with n^2 is a square.

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A075404 Smallest m > n such that Sum_{i=n..m} i^2 is a square, or 0 if no such m exists.

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A184762 The number of numbers k > n such that Sum_{i=n..k} i^2 is a square.

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A180259 Squares which are the sum of consecutive squares starting with 25^2.

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A180273 Squares which are a sum of consecutive squares starting with 7^2.

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A184885 Irregular triangle in which row n has the values of k>1 such that sum_{i=n..n+k-1} i^2 is a square.

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A180465 Squares which are a sum of consecutive squares starting with 38^2.

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