A063014 -id:A063014 - cp's OEIS frontend

A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, 12, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 4, 20, 12, 4, 4, 12, 12, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 12, 4, 4, 4, 12, 4, 4, 4, 4, 20, 12, 12, 12, 4, 12, 4, 4, 12, 4, 12, 12, 4, 4, 4, 36, 4, 4, 12, 4, 12, 4, 4, 12, 12, 20, 4, 4, 12, 4, 12, 4, 12, 4, 4, 36

Offset: 0

Eric W. Weisstein

Also number of Gaussian integers x + yi having absolute value n. - Alonso del Arte, Feb 11 2012

The indices of terms not equaling 4 or 12 correspond to A009177, n>0. - Bill McEachen, Aug 14 2025

			a(5) = 12 because the circumference of the circle with radius 5 will pass through the twelve points (5, 0), (4, 3), (3, 4), (0, 5), (-3, 4), (-4, 3), (-5, 0), (-4, -3), (-3, -4), (0, -5), (3, -4) and (4, -3). Alternatively, we can say the twelve Gaussian integers 5, 4 + 3i, ... , 4 - 3i all have absolute value of 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael Gilleland, Some Self-Similar Integer Sequences
Eric Weisstein's World of Mathematics, Circle Lattice Points

Cf. A004018, A046080, A046110, A046111, A046112, A063014, A256452.

Cf. A000328, A051132, A009177.

a046109 n = length [(x,y) | x <- [-n..n], y <- [-n..n], x^2 + y^2 == n^2]
-- Reinhard Zumkeller, Jan 23 2012

N:= 1000: # to get a(0) to a(N)
A:= Array(0..N):
A[0]:= 1:
for x from 1 to N do
  A[x]:= A[x]+4;
  for y from 1 to min(x-1,floor(sqrt(N^2-x^2))) do
     z:= x^2+y^2;
     if issqr(z) then
       t:= sqrt(z);
       A[t]:= A[t]+8;
     fi
  od
od:
seq(A[i],i=0..N); # Robert Israel, May 08 2015

Table[Length[Select[Flatten[Table[r + I i, {r, -n, n}, {i, -n, n}]], Abs[#] == n &]], {n, 0, 49}] (* Alonso del Arte, Feb 11 2012 *)

a(n)=if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, 2*f[i,2]+1, 1)) \\ Charles R Greathouse IV, Feb 01 2017

a(n)=if(n==0, return(1)); t=0; for(x=1, n-1, y=n^2-x^2; if(issquare(y), t++)); return(4*t+4) \\ Arkadiusz Wesolowski, Nov 14 2017

from sympy import factorint
def a(n):
    r = 1
    for p, e in factorint(n).items():
        if p%4 == 1: r *= 2*e + 1
    return 4*r if n > 0 else 0
# Orson R. L. Peters, Jan 31 2017

a(n) = A000328(n) - A051132(n).
a(n) = 8*A046080(n) + 4 for n > 0.
a(n) = A004018(n^2).
From Jean-Christophe Hervé, Dec 01 2013: (Start)
a(A084647(k)) = 28.
a(A084648(k)) = 36.
a(A084649(k)) = 44. (End)
a(n) = 4 * Product_{i=1..k} (2*e_i + 1) for n > 0, given that p_i^e_i is the i-th factor of n with p_i = 1 mod 4. - Orson R. L. Peters, Jan 31 2017
a(n) = [x^(n^2)] theta_3(x)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018
From Hugo Pfoertner, Sep 21 2023: (Start)
a(n) = 8*A063014(n) - 4 for n > 0.
a(n) = 4*A256452(n) for n > 0. (End)

A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 1, 2, 2, 1, 2, 1, 0, 1, 1, 2, 2, 2, 2, 1, 1, 0, 1, 1, 2, 3, 3, 3, 2, 1, 1, 0, 1, 1, 2, 3, 3, 4, 4, 2, 1, 1, 0, 1, 1, 2, 3, 4, 5, 5, 4, 1, 1, 1, 0, 1, 1, 2, 4, 5, 7, 9, 6, 2, 4, 2, 1, 0

Offset: 0

Alois P. Heinz, Feb 17 2015

			Square array A(n,k) begins:
  1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1, ...
  0, 1, 1, 1, 2,  2,  2,  2,  2,  2,  2, ...
  0, 1, 1, 2, 2,  2,  3,  3,  3,  4,  4, ...
  0, 1, 1, 1, 2,  3,  3,  4,  5,  5,  6, ...
  0, 1, 2, 2, 3,  4,  5,  7,  8,  9, 11, ...
  0, 1, 1, 2, 4,  5,  9, 10, 11, 15, 17, ...
  0, 1, 1, 2, 4,  6,  9, 13, 18, 21, 27, ...
  0, 1, 1, 1, 2,  7,  9, 16, 25, 30, 41, ...
  0, 1, 1, 4, 6,  8, 18, 27, 36, 52, 68, ...
  0, 1, 2, 2, 7, 13, 23, 36, 51, 70, 94, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000007, A000012, A063014, A016727, A065458, A065459, A065460, A065461, A065462, A255213, A255214.
Main diagonal gives A105152.
Cf. A302996.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1 and n<=t, 1,
      (j-> `if`(t*jn, 0, b(n-j, i, t-1))))(i^2))
    end:
A:= (n, k)-> b(n^2, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1 && n <= t, 1, Function[j, If[t*jn, 0, b[n-j, i, t-1]]]][i^2]]; A[n_, k_] := b[n^2, n, k]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A286361 Least number with the same prime signature as {the largest divisor of n with only prime factors of the form 4k+1} has: a(n) = A046523(A170818(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 6, 1, 1, 2, 1, 2, 1, 1, 2, 2, 4, 1, 1, 2, 1, 2, 1, 2, 1, 1, 6, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2

Offset: 1

Antti Karttunen, May 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A170818, A267099, A267113, A286363, A286364, A286365.

Differs from A063014 for the first time at n=25, where a(25) = 4, while A063014(25) = 3.

from sympy import factorint
from operator import mul
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a072438(n):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==1 else i**f[i] for i in f])
def a(n): return a046523(n/a072438(n)) # Indranil Ghosh, May 09 2017

(define (A286361 n) (A046523 (A170818 n)))

a(n) = A046523(A170818(n)).

a(n) = A286363(A267099(n)).

A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 2, 6, 7, 6, 4, 8, 10, 14, 2, 11, 14, 13, 7, 23, 15, 17, 4, 24, 21, 31, 10, 25, 37, 28, 2, 46, 29, 49, 14, 38, 35, 61, 7, 45, 62, 49, 15, 93, 47, 57, 4, 72, 67, 97, 21, 71, 84, 101, 10, 119, 70, 86, 37, 92, 79, 165, 2, 138, 127, 109, 29, 168, 140, 121, 14

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=3 because 25 produces {0,0,0,5}, {0,0,3,4}, {1,2,2,4}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A063014, A016727, A065459.

Column k=4 of A255212.

N:= 100:
R:= Vector(N,1):
for a from 0 to N do
  for b from a to floor(sqrt(N^2-a^2)) do
     for c from b to floor(sqrt(N^2-a^2-b^2)) do
       q:= a^2 + b^2 + c^2;
       for f in numtheory:-divisors(q) do
          if f^2 + 2*f*c <= q and (f + q/f)::even then
             r:= (f + q/f)/2;
             if r <= N then R[r]:= R[r]+1 fi;
          fi
od od od od:
convert(R,list); # Robert Israel, Feb 16 2015

Length/@Table[SumOfSquaresRepresentations[4, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065459 Number of inequivalent (ordered) solutions to n^2 = sum of 5 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 13, 12, 13, 17, 25, 22, 27, 31, 35, 38, 46, 49, 61, 61, 61, 73, 92, 83, 112, 106, 118, 127, 147, 138, 185, 175, 178, 198, 239, 212, 254, 262, 298, 294, 341, 304, 404, 376, 385, 432, 483, 441, 539, 517, 560, 551, 680, 587, 745, 693, 698

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=4 because 25 produces {0,0,0,0,5}, {0,0,0,3,4}, {0,1,2,2,4}, {2,2,2,2,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..750

Cf. A063014, A016727.

Column k=5 of A255212.

Length/@Table[SumOfSquaresRepresentations[5, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065461 Number of inequivalent (ordered) solutions to n^2 = sum of 7 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 13, 16, 27, 36, 43, 58, 72, 99, 130, 146, 178, 254, 265, 342, 417, 507, 540, 726, 745, 975, 1092, 1289, 1338, 1845, 1751, 2246, 2447, 2948, 2852, 3932, 3638, 4728, 4868, 5778, 5618, 7659, 6887, 8891, 8887, 10825, 10109, 13712

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(4)=4 because 16 produces {0,0,0,0,0,0,4}, {0,0,0,2,2,2,2}, {0,0,1,1,1,2,3}, {1,1,1,1,2,2,2}.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A063014, A016727.

Column k=7 of A255212.

Table[Length[PowersRepresentations[n^2,7,2]],{n,0,50}] (* Harvey P. Dale, Sep 08 2020 *)

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

Previous Mathematica program replaced by Harvey P. Dale, Sep 08 2020

A065460 Number of inequivalent (ordered) solutions to n^2 = sum of 6 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 9, 9, 9, 18, 23, 24, 29, 37, 53, 62, 59, 77, 116, 106, 130, 156, 199, 192, 221, 257, 342, 336, 384, 402, 577, 501, 599, 639, 835, 774, 910, 912, 1220, 1113, 1378, 1298, 1703, 1530, 1907, 1862, 2398, 2094, 2471, 2393, 3356, 2765, 3543

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(5)=5 because 25 produces {0,0,0,0,0,5}, {0,0,0,0,3,4}, {0,0,1,2,2,4}, {0,2,2,2,2,3}, {1,1,1,2,3,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A063014, A016727.

Column k=6 of A255212.

Length/@Table[SumOfSquaresRepresentations[6, (k)^2], {k, 72}]

a(0)=1 prepended by Alois P. Heinz, Feb 17 2015

A065462 Number of inequivalent (ordered) solutions to n^2 = sum of 8 squares of integers >= 0.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 18, 25, 36, 51, 73, 90, 133, 169, 223, 295, 380, 452, 603, 763, 903, 1115, 1385, 1668, 2025, 2398, 2811, 3535, 4011, 4683, 5503, 6724, 7316, 8684, 9946, 11844, 12994, 15091, 16712, 20493, 21663, 24975, 27536, 33079, 34654, 39957, 43315

Offset: 0

Wouter Meeussen, Nov 18 2001

nonn

			a(4)=5 because 16 produces {0,0,0,0,0,0,0,4}, {0,0,0,0,2,2,2,2}, {0,0,0,1,1,1,2,3}, {0,1,1,1,1,2,2,2}, {1,1,1,1,1,1,1,3}.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A016727, A063014.

Column k=8 of A255212.

Length/@Table[SumOfSquaresRepresentations[8, (k)^2], {k, 36}]

a(0), a(37)-a(47) from Alois P. Heinz, Feb 16 2015

A063669 Hypotenuses of reciprocal Pythagorean triangles: number of solutions to 1/(12n)^2 = 1/b^2 + 1/c^2 [with b >= c > 0]; also number of values of A020885 (with repetitions) which divide n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5

Offset: 1

Henry Bottomley, Jul 28 2001

nonn

Primitive reciprocal Pythagorean triangles 1/a^2 = 1/b^2 + 1/c^2 have a=fg, b=ef, c=eg where e^2 = f^2 + g^2; i.e., e,f,g represent the sides of primitive Pythagorean triangles. But the product of the two legs of primitive Pythagorean triangles are multiples of 12 and so the reciprocal of hypotenuses of reciprocal Pythagorean triangles are always multiples of 12 (A008594).

			a(1)=1 since 1/(12*1)^2 = 1/12^2 = 1/15^2 + 1/20^2;
a(70)=6 since 1/(12*70)^2 = 1/840^2 = 1/875^2 + 1/3000^2 = 1/888^2 + 1/2590^2 = 1/910^2 + 1/2184^2 = 1/952^2 + 1/1785^2 = 1/1050^2 + 1/1400^2 = 1/1160^2 + 1/1218^2.
Looking at A020885, 1 is divisible by 1, while 70 is divisible by 1, 5, 10, 14, 35 and again 35.

Cf. A046080, A063664, A063665, A063014.

cp's OEIS Frontend

A046109 Number of lattice points (x,y) on the circumference of a circle of radius n with center at (0,0).

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A255212 Number A(n,k) of partitions of n^2 into at most k square parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A286361 Least number with the same prime signature as {the largest divisor of n with only prime factors of the form 4k+1} has: a(n) = A046523(A170818(n)).

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A065458 Number of inequivalent (ordered) solutions to a^2 + b^2 + c^2 + d^2 = n^2.

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A065459 Number of inequivalent (ordered) solutions to n^2 = sum of 5 squares of integers >= 0.

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A065461 Number of inequivalent (ordered) solutions to n^2 = sum of 7 squares of integers >= 0.

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A065460 Number of inequivalent (ordered) solutions to n^2 = sum of 6 squares of integers >= 0.

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A065462 Number of inequivalent (ordered) solutions to n^2 = sum of 8 squares of integers >= 0.

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