A062861 -id:A062861 - cp's OEIS frontend

A212016 Sums of the squares of two or more consecutive integers.

1, 2, 5, 6, 10, 13, 14, 15, 19, 25, 28, 29, 30, 31, 35, 41, 44, 50, 54, 55, 56, 60, 61, 69, 77, 85, 86, 90, 91, 92, 96, 105, 110, 113, 121, 126, 135, 139, 140, 141, 145, 146, 149, 154, 170, 174, 181, 182, 190, 194, 195, 199, 203, 204, 205, 209, 218

Offset: 1

Max Alekseyev, Apr 26 2012

Subsequence of A062861.
Contains A212015 as a subsequence.
A174069 is a subsequence. - Altug Alkan, Dec 24 2015

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062861, A174069, A212015.

N:= 1000: # to get all terms <=N
g:= x -> x*(x+1)*(2*x+1)/6:
S:= select(`<=`,{seq(seq(g(b)-g(a), a= -b-1 .. b-2), b = 1..floor((sqrt(2*N-1)+1)/2))},N):
sort(convert(S,list)); # Robert Israel, Jan 05 2016

{ isA212016(t) = fordiv(6*t,k, if(k==1,next); z=(k^2-1)/3; if(issquare(4*t/k-z), return(k)); if(z>4*t/k,break); ); 0 }

A212015 Nonsquare positive integers that are sums of consecutive integer squares.

Original entry on oeis.org

2, 5, 6, 10, 13, 14, 15, 19, 28, 29, 30, 31, 35, 41, 44, 50, 54, 55, 56, 60, 61, 69, 77, 85, 86, 90, 91, 92, 96, 105, 110, 113, 126, 135, 139, 140, 141, 145, 146, 149, 154, 170, 174, 181, 182, 190, 194, 195, 199, 203, 204, 205, 209, 218, 221, 230, 231

Offset: 1

Max Alekseyev, Apr 26 2012

nonn

Nonsquare terms of A062861.

Subsequence of A212016.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062861, A212016.

filter:= proc(n)
  not issqr(n) and
  ormap(k -> issqr(-3*k^4+3*k^2+36*k*n) and  ((3*k-3*k^2+sqrt(-3*k^4+3*k^2+36*k*n))/(6*k))::integer,
    numtheory:-divisors(6*n))
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 22 2017

filterQ[n_] := !IntegerQ[Sqrt[n]] && AnyTrue[Divisors[6n], IntegerQ[Sqrt[-3 #^4 + 3 #^2 + 36 # n]] && IntegerQ[(3 # - 3 #^2 + Sqrt[-3 #^4 + 3 #^2 + 36 # n])/(6#)]&];
Select[Range[1000], filterQ] (* Jean-François Alcover, Jun 08 2020, after Maple *)

A062862 Number of ways n can be written as the sum of squares of consecutive numbers (possibly including squares of negative numbers).

Original entry on oeis.org

1, 4, 1, 0, 2, 4, 2, 0, 0, 2, 1, 0, 0, 2, 4, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, 4, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 4, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 0, 2, 4, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0

Offset: 0

Henry Bottomley, Jun 25 2001

nonn

			a(25)=4 since 25 = 5^2 = 3^2+4^2 = (-4)^2+(-3)^2 = (-5)^2.

Index entries for sequences related to sums of squares

If a(n) is zero then n is in A062863, otherwise in A062861. If a(n) is odd then n is in A006331.

A062863 Numbers which cannot be written as sums of squares of consecutive numbers (even allowing squares of negative numbers).

Original entry on oeis.org

3, 7, 8, 11, 12, 17, 18, 20, 21, 22, 23, 24, 26, 27, 32, 33, 34, 37, 38, 39, 40, 42, 43, 45, 46, 47, 48, 51, 52, 53, 57, 58, 59, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 82, 83, 84, 87, 88, 89, 93, 94, 95, 97, 98, 99, 101, 102, 103, 104, 106, 107, 108

Offset: 0

Henry Bottomley, Jun 25 2001

nonn

Index entries for sequences related to sums of squares

Complement of A062861. Cf. A062862.

Complement[Range[150],Union[Flatten[Table[Union[Total/@(Partition[ Range[-50,50],i,1]^2)],{i,20}]]]] (* Harvey P. Dale, Jan 04 2012 *)

A212017 Integers n such that n^3 is the sum of consecutive integer squares.

Original entry on oeis.org

0, 1, 4, 9, 11, 16, 25, 26, 36, 47, 49, 64, 65, 66, 81, 100, 109, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 921, 935, 961, 1024, 1079, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600

Offset: 1

Max Alekseyev, Apr 26 2012

nonn

Integers n such that n^3 belongs to A062861.

A212018 Integers m such that m^3 is the sum of two or more consecutive integer squares.

Original entry on oeis.org

1, 11, 26, 47, 65, 66, 109, 921, 935, 1079, 2161, 2820, 2860, 5029, 9105, 10681, 12284, 13156, 16761, 18340, 41921, 43500, 61721, 63765, 64605, 66317, 75130, 99359, 105731, 116180, 122009, 146821, 159371, 218205, 253393, 260165, 264680, 269588, 314919, 403130, 404326, 420365

Offset: 1

Max Alekseyev, Apr 26 2012

nonn

Integers m such that m^3 belongs to A212016.

Are there any squares besides 1 in this sequence?

Xianwen Wang, Table of n, a(n) for n = 1..146 (all terms up to 10^10).

Subsequence of A212017.

Cf. A062861, A212015, A212016.

a(39)-a(42) from Xianwen Wang, May 23 2025

A182379 Positive integers n such that n^2 is the sum of two or more consecutive integer squares.

Original entry on oeis.org

1, 5, 11, 29, 34, 38, 39, 50, 55, 70, 77, 88, 92, 94, 105, 106, 115, 135, 138, 143, 155, 158, 169, 182, 185, 189, 195, 242, 245, 253, 274, 284, 316, 321, 332, 356, 357, 385, 413, 430, 440, 495, 511, 531, 650, 652, 655, 671, 676, 679, 724, 726, 764

Offset: 1

Max Alekseyev, Apr 26 2012

nonn

Integers n such that n^2 belongs to A212016.

{ a(n)^2 : n=1,2,... } forms the set difference of A212016 and A212015.

Cf. A062861, A212017, A212018.

cp's OEIS Frontend

A212016 Sums of the squares of two or more consecutive integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A212015 Nonsquare positive integers that are sums of consecutive integer squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A062862 Number of ways n can be written as the sum of squares of consecutive numbers (possibly including squares of negative numbers).

Views

Author

Keywords

Examples

Links

Crossrefs

A062863 Numbers which cannot be written as sums of squares of consecutive numbers (even allowing squares of negative numbers).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A212017 Integers n such that n^3 is the sum of consecutive integer squares.

Views

Author

Keywords

Comments

A212018 Integers m such that m^3 is the sum of two or more consecutive integer squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A182379 Positive integers n such that n^2 is the sum of two or more consecutive integer squares.

Views

Author

Keywords

Comments

Crossrefs