A383359
Integers m such that m^4 is the sum of squares of two or more consecutive positive integers.
Original entry on oeis.org
13, 295, 330, 364, 1085, 5005, 6305, 15516, 415151, 1990368, 34011252, 42016497, 79565281, 139107722, 254801664, 418093065, 667378972, 1214995500, 3609736702, 4353556896
Offset: 1
295 is a term because 295^4 = 6453^2 + 6454^2 + ... + 6628^2 + 6629^2.
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d[m_] := Select[Divisors[6 m^4], 1 < # < Floor@ CubeRoot[3 m^4] &&
IntegerQ[1/6 (-3 (1 + #) + Sqrt[3 (12 m^4/# + 1 - #^2)])] &];
Do[If[Length@d[m] > 0, Print[m]], {m, 1, 10000}]
A383367
a(n) is the least integer k such that A383359(n)^4 can be expressed as a sum of squares of k consecutive integers.
Original entry on oeis.org
2, 177, 352, 1536, 2401, 40898, 60625, 185761, 19512097, 47761921, 1224370081, 7957888849, 10842382346, 11474926944, 12230369281, 190412616875, 497818686976, 72899460001, 1384334025217, 313455536641
Offset: 1
a(2) = 177 because A383359(2)^4 = 295^4 can be expressed as sum of squares of 177 consecutive integers: 295^4 = 6453^2 + 6454^2 + ... + 6628^2 + 6629^2.
A383653
Integers m such that m^4 is the sum of squares of two or more consecutive integers, positive or negative.
Original entry on oeis.org
1, 13, 26, 33, 295, 330, 364, 1085, 5005, 5546, 5682, 6305, 6538, 15516, 415151, 1990368, 3538366, 34011252, 42016497, 79565281, 139107722, 175761059, 254801664, 418093065, 667378972, 1214995500, 3609736702, 4353556896
Offset: 1
5546 is a term because 5546^4 = (-22205)^2 + (-22204)^2 + ... + 141400^2 + 141401^2.
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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
A383654
a(n) is the number k such that A383653(n)^4 is the sum of squares of k consecutive integers.
Original entry on oeis.org
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
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.
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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
Showing 1-4 of 4 results.
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