A383654 a(n) is the number k such that A383653(n)^4 is the sum of squares of k consecutive integers.

2, 2, 169, 242, 177, 352, 1536, 2401, 40898, 163607, 230121, 60625, 218089, 185761, 19512097, 47761921, 1170329056, 1224370081, 7957888849, 10842382346, 11474926944, 208152552417, 12230369281, 190412616875, 497818686976, 72899460001, 1384334025217, 313455536641

Offset

1,1

Links

Table of n, a(n) for n=1..28.

Zhining Yang, Can be expressed as the fourth power of the sum of squares of consecutive positive integers, Chinese BBS.

Example

Case a(1)=2: 13^4 = 119^2 + 120^2, 1^4 = 0^2 + 1^2.
Case a(3)=169: 26^4 = (-67+1)^2 + (-67+2)^2 + ... + (-67+168)^2 + (-67+169)^2.
Case a(5)=177: 295^4 = (6452+1)^2 + (6452+2)^2 + ... + (6452+176)^2 + (6452+177)^2.
...
Case a(10)=163607: 5546^4 = (-22206+1)^2 + (-22206+2)^2 + ... + (-22206+163606)^2 + (-22206+163607)^2.

Mathematica

lst={}; Monitor[Do[mm=6 m^4; div=TakeWhile[Divisors[mm][[2;; -2]], 2mm/#+1>#^2&];
ans=Select[div, IntegerQ[Sqrt[(2mm/#+1-#^2)/3]]&&Mod[#-Sqrt[(2mm/#+1-#^2)/3], 2]==1&];
If[Length[ans]>0, tmp={m, {#, q=Sqrt[(2mm/#+1-#^2)/3], p=(q+1-#)/2}&/@ans}; Print[tmp];
AppendTo[lst, tmp]], {m, 1, 10^4}], m]; lst

Crossrefs

Cf. A000330, A097812, A189173, A383359, A383367, A383653.

Sequence in context: A395697 A100956 A358621 * A304915 A216481 A078241

Adjacent sequences: A383651 A383652 A383653 * A383655 A383656 A383657

Keyword

nonn,more

Author

Xianwen Wang, May 04 2025

Status

approved