A180442 Numbers n such that a sum of two or more consecutive squares beginning with n^2 is a square.
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
Examples
30 is in the sequence because 30^2 + 31^2 + 32^2 + ... + 197^2 + 198^2 = 1612^2.
Links
- 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
Programs
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Mathematica
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 *)
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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())))
Formula
Numbers n such that A075404(n) > 0.
Extensions
Example simplified by Jon E. Schoenfield, Sep 18 2013
More terms from Jon E. Schoenfield, Nov 22 2013
Comments