A054994 -id:A054994 - cp's OEIS frontend

A097756 Table read by rows of A054994 ordered by A046080.

1, 5, 25, 125, 65, 625, 3125, 15625, 325, 78125, 390625, 1953125, 1625, 9765625, 48828125, 4225, 244140625, 1105, 8125, 1220703125, 6103515625, 30517578125, 40625, 152587890625, 21125, 762939453125, 3814697265625, 203125

Offset: 1

Ray Chandler, Aug 26 2004

Row n of table is A054994(k) such that A046080(A054994(k)) = n; number of terms in row n is A001055(2n+1).

Column 1 of table gives A006339 (or A046112).

			Table begins:
0: 1,
1: 5,
2: 25,
3: 125,
4: 65,625,
5: 3125,
6: 15625,
7: 325,78125,
8: 390625,
9: 1953125,
10: 1625,9765625,
11: 48828125,
12: 4225,244140625,
13: 1105,8125,1220703125,
14: 6103515625,
15: 30517578125,
16: 40625,152587890625,
17: 21125,762939453125,
18: 3814697265625,
19: 203125,19073486328125,
20: 95367431640625,
...

Ray Chandler, Table of n, a(n) for n = 1..4522, 1431 rows flattened.
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A097751, A097752, A097753, A097754, A097755, A054994.

A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.

Original entry on oeis.org

0, 1, 5, 25, 65, 325, 1105, 4225, 5525, 27625, 71825, 138125, 160225, 801125, 2082925, 4005625, 5928325, 29641625, 77068225, 148208125, 243061325, 1215306625, 3159797225, 6076533125, 12882250225, 53716552825, 64411251125

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

The number of lattice points (i,j) on the circle with i^2 + j^2 = a(n) is given by A071385(n).
In a sci.math posting on May 05 2002 entitled "Circle with 3 lattice points", James R. Buddenhagen asked: Which circles have the property that they pass through more lattice points than any smaller circle? and he gave the terms 1, 25, 65, 325, 1105, 4225, 5525, with the missing 5 added by Ahmed Fares. In the same thread Gerry Myerson mentioned the factorization into primes of the form 4*k+1.
Also, numbers with a record number of divisors all of whose prime factors are of the form 4k + 1. - Amiram Eldar, Sep 12 2019
Indices of records of A004018. Apart from the first term, also indices of records of A002654. - Jianing Song, May 20 2021

Ray Chandler, Table of n, a(n) for n = 1..425 (first 97 terms from Ray Chandler, terms 98-365 from Amiram Eldar)
James Buddenhagen, Circle with 3 lattice points, thread in sci.math (May 2002)
Hugo Pfoertner, Construction of the sequences A071383, A071384, A071385

Cf. A000448, A048610, A052199, A071384, A071385, A230655, A300162. Subsequence of A054994 (excluding first term). Where records occur in A004018. See A088959 for circles with integer radius.

Indices of records of Sum_{d|n} kronecker(m, d): A230655 (m=-3), this sequence (m=-4), A279541 (m=-6).

my(v=list(10^15), rec=0); print1(0, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(v[n], ", "))) \\ Jianing Song, May 20 2021, see program for A054994

from math import prod
from sympy import isprime
primes_congruent_1_mod_4 = [5]
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
    while i>=len(primes_congruent_1_mod_4): # generate primes on demand
        n = primes_congruent_1_mod_4[-1]+4
        while not isprime(n): n += 4
        primes_congruent_1_mod_4.append(n)
    return primes_congruent_1_mod_4[i]
def generate_A054994():
    TO_DO = {(1,())}
    while True:
        radius, exponents = min(TO_DO)
        yield radius, exponents
        TO_DO.remove((radius, exponents))
        TO_DO.update(successors(radius,exponents))
def successors(r,exponents):
    for i,e in enumerate(exponents):
        if i==0 or exponents[i-1]>e:
            yield (r*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
    if exponents==() or exponents[-1]>0:
        yield (r*prime_4k_plus_1(len(exponents)), exponents+(1,))
n,record,radius=1,1,0
print(radius, end="") # or record, for A071385
for radius,expo in generate_A054994():
    num_points = 4*prod((e+1) for e in expo)
    if num_points>record:
        record = num_points
        n += 1
        print (",", radius, end="") # or record, for A071385
        if n==27: break
print()
# Günter Rote, Sep 12 2023

For n>1 we have 1 < a(n+1)/a(n) <= 5, since one can multiply the points x+iy for which x^2 + y^2 = N by either 2+i or 2-i to get two new sets of points X+iY for which X^2 + Y^2 = 5N. This strictly increases the number since it is easy to see that the two sets aren't the same. - J. H. Conway, Jun 04 2002

lim n ->infinity Log(a(n))/n = 1. [Conjectured by Benoit Cloitre, proved by J. H. Conway]

A093195 Least number which is the sum of two distinct nonzero squares in exactly n ways.

Original entry on oeis.org

5, 65, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 126953125, 160225, 1221025, 3453125, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 30994415283203125, 5928325, 303460625, 53955078125, 35409725, 100140625

Offset: 1

Lekraj Beedassy, Apr 22 2004

nonn

An algorithm to compute the n-th term of this sequence: Write each of 2n and 2n+1 as products of their divisors in all possible ways and in decreasing order. For each product, equate each divisor in the product to (a1+1)(a2+1)...(ar+1), so that a1 >= a2 >= a3 >= ... >= ar, and solve for the ai. Evaluate A002144(1)^a1 * A002144(2)^a2 * ... * A002144(r)^ar for each set of values determined above, then the smaller of these products is the least integer to have precisely n partitions into a sum of two distinct positive squares. [Ant King, Dec 14 2009; May 26 2010]

Ray Chandler, Table of n, a(n) for n = 1..1438 (a(1439) exceeds 1000 digits).

Cf. A002144, A018782, A054994, A025302-A025311 (first entries). See A016032, A000446 and A124980 for other versions.

b(k)=my(c=0);for(i=1,sqrtint((k-1)\2),if(issquare(k-i^2),c+=1));c \\ A025441
for(n=1,10,k=1;while(k,if(b(k)==n,print1(k,", ");break);k+=1)) \\ Derek Orr, Mar 20 2019

a(n) = min(A018782(2n), A018782(2n+1)).

More terms from Ant King, Dec 14 2009 and Feb 07 2010

A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

Original entry on oeis.org

1, 5, 25, 65, 625, 325, 15625, 1105, 4225, 8125, 9765625, 5525, 244140625, 203125, 105625, 27625, 152587890625, 71825, 3814697265625, 138125, 2640625, 126953125, 2384185791015625, 160225, 17850625, 3173828125, 1221025, 3453125

Offset: 1

N. J. A. Sloane

nonn

a(n) is least term of A054994 with exactly n divisors. - Ray Chandler, Jan 05 2012
From Jianing Song, Apr 24 2021: (Start)
a(n) is the smallest k such that A004018(k) = 4n.
Also a(n) is the smallest index of n in A002654.
a(n) is the smallest term in A004613 that has exactly n divisors.
This is a subsequence of A054994. (End)

			4225 = 5^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 4 with exactly 9 divisors, so a(9) = 4225. - _Jianing Song_, Apr 24 2021

Ray Chandler, Table of n, a(n) for n = 1..1432 (a(1433) exceeds 1000 digits).

Cf. A002144, A002654, A004018, A005179, A054994, A000446, A124980, A016032, A093195.

(* This program is not convenient to compute huge terms - A054994 is assumed to be computed with maxTerm = 10^16 *) a[n_] := Catch[ For[k = 1, k <= Length[A054994], k++, If[DivisorSigma[0, A054994[[k]]] == n, Throw[A054994[[k]]]]]]; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jan 21 2013, after Ray Chandler *)

primelist(d,r,l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p,d)==r, i++; v[i]=p; if(i==l, break())))); v
prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
A018782(n)=my(pf=prodR(n, n), a=1, b, v=primelist(4, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(bJianing Song, Apr 25 2021, following R. J. Mathar's program for A005179.

A000446(n) = min(a(2n-1), a(2n)) for n > 1.
A124980(n) = min(a(2n-1), a(2n)).
A016032(n) = min(2*a(2n-1), a(2n), a(2n+1)).
A093195(n) = min(a(2n), a(2n+1)).
From Jianing Song, Apr 24 2021: (Start)
If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 4. The smallest n such that the equality does not hold is n = 16.
a(n) <= 5^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
a(p*q) = 5^(p-1) * 13^(q-1) for primes p >= q. (End)

A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, 640, 768, 864, 1024, 1152, 1280, 1536, 1728, 2048, 2304, 2560, 3072, 3456, 3840, 4096, 4608, 5120, 6144, 6912, 7680, 8192, 9216, 10240, 11520, 12288, 13824, 15360

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

			Circles with radius 1 and 2 have 4 lattice points on their circumference, so a(1)=4. A circle with radius sqrt(5) passes through 8 lattice points of the shape (2,1), so a(2)=8. A circle with radius 5 passes through 4 lattice points of shape (5,0) and through 8 points of shape (4,3), so a(3)=4+8=12
A071383(11) = 5^2 * 13^2 * 17^1 = 71825. Therefore A071385(11) = 4*(2+1)*(2+1)*(1+1) = 72.

Ray Chandler, Table of n, a(n) for n = 1..425
Hugo Pfoertner, Construction of A071383, A071384, A071385

Records in A004018. Cf. A071383, A071384, A002144, A002654, A300163, A344470.

my(v=list(10^15), rec=0); print1(1, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(4*rec, ", "))) \\ Jianing Song, May 20 2021, see program for A054994

a(n) = 4 * Product_{k=1..klim} (e_k + 1), where klim and e_1 >= e_2 >= ... >= e_klim > 0 are known from A071383(n) = Product_{k=1..klim} p_k^e_k, with p_k = k-th prime of the form 4i+1. (J. H. Conway)
a(n) = 4*A000005(A071383(n)) for n > 1.
a(n) = A004018(A071383(n)).
a(n) = A002654(A071383(n)) for n > 1. - Jianing Song, May 20 2021
a(n) = 4*A344470(n-1) for n > 1. - Hugo Pfoertner, Sep 04 2022

A000446 Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

Original entry on oeis.org

0, 25, 325, 1105, 4225, 5525, 203125, 27625, 71825, 138125, 2640625, 160225, 17850625, 1221025, 1795625, 801125, 1650390625, 2082925, 49591064453125, 4005625, 44890625, 2158203125, 30525625, 5928325, 303460625, 53955078125

Offset: 1

N. J. A. Sloane

nonn

Could start with a(0) = 3: the smallest nonnegative integer that can be written as sum of two squares in 0 ways. - M. F. Hasler, Jul 05 2024

			a(1) = 0 because 0 is the smallest integer which is uniquely a unique sum of two squares, namely 0^2 + 0^2.
a(2) = 25 from 25 = 5^2 + 0^2 = 3^2 + 4^2.
a(3) = 325 from 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(4) = 1105 from 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.

Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A000448 (similar, but "in at least n ways").
Cf. A002144, A018782, A054994.
See A016032, A093195 and A124980 for other versions.

A000446(n)=if(n>1, A124980(n), 0) \\ M. F. Hasler, Jul 06 2024

A000446=lambda n: A124980(n) if n>1 else 0 # M. F. Hasler, Jul 07 2024

An algorithm to compute the n-th term of this sequence for n>1: Write each of 2n and 2n-1 as products of their divisors, in decreasing order and in all possible ways. Equate each divisor in the product to (a1+1)(a2+1)...(ar+1), so that a1>=a2>=a3>=...>=ar, and solve for the ai. Evaluate A002144(1)^a1 x A002144(2)^a2 x ... x A002144(r)^ar for each set of values determined above, then the smaller of these products is the least integer to have precisely n partitions into a sum of two squares. [Ant King, Oct 07 2010]
a(n) = min(A018782(2n-1), A018782(2n)) for n > 1.
a(n) = A124980(n) for n > 1. - M. F. Hasler, Jul 07 2024

Better description and more terms from David W. Wilson, Aug 15 1996

Definition improved by several correspondents, Nov 12 2007

A052199 Numbers that are expressible as the sum of 2 distinct positive squares in more ways than any smaller number.

Original entry on oeis.org

1, 5, 65, 325, 1105, 5525, 27625, 71825, 138125, 160225, 801125, 2082925, 4005625, 5928325, 29641625, 77068225, 148208125, 243061325, 1215306625, 3159797225, 6076533125, 12882250225, 53716552825, 64411251125, 167469252925, 322056255625, 785817263725

Offset: 1

Jud McCranie, Jan 28 2000

nonn

			65 = 1^2 + 8^2 = 4^2 + 7^2, the smallest expressible in two ways, so 65 is a term.

Donald S. McDonald, Postings to sci.math newsgroup, Feb 21, 1995 and Dec 04, 1995.

Ray Chandler, Table of n, a(n) for n = 1..422
Dirk Frettlöh, "Tile Orientations with Distinct Frequencies", Ch. 1.5 in Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity, 2017, see page 9.
Donald S. McDonald, World Mathematical Year 2000 Poster Competition
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A001983, A007511, A048610, A071383. Subsequence of A054994. Where records occur in A025441; corresponding number of ways is A060306.

c_old=-1;for(n=1,10000,c=0;for(i=1,floor(sqrt(n)),for(j=1,i-1,if(i^2+j^2==n,c+=1)));if(c>c_old,print1(n,", ");c_old=c)) \\ Derek Orr, Mar 15 2019

More terms from Randall L Rathbun, Jan 18 2002

Edited by Ray Chandler, Jan 12 2012

A088959 Lowest numbers which are d-Pythagorean decomposable, i.e., square is expressible as sum of two positive squares in more ways than for any smaller number.

Original entry on oeis.org

1, 5, 25, 65, 325, 1105, 5525, 27625, 32045, 160225, 801125, 1185665, 5928325, 29641625, 48612265, 243061325, 1215306625, 2576450045, 12882250225, 64411251125, 157163452745, 785817263725, 3929086318625, 10215624428425, 11472932050385, 51078122142125

Offset: 1

Lekraj Beedassy, Dec 01 2003

nonn

These are also the integer radii of circles around the origin that contain record numbers of lattice points. See A071383 for radii that are not necessarily integer. - Günter Rote, Sep 14 2023

			From _Petros Hadjicostas_, Jul 21 2019: (Start)
Squares 1^2, 2^2, 3^2, and 4^2 have 0 representations as the sum of two positive squares. (Thus, A088111(1) = 0 for the number of representations of 1^2.) Thus a(1) = 1.
Square 5^2 can be written as 3^2 + 4^2 only (here A088111(2) = 1). Thus, a(2) = 5.
Looking at sequence A046080, we see that for 5 <= n <= 24, only n^2 = 5^2, 10^2, 13^2, 15^2, 17^2, 20^2 can be written as a sum of two positive squares (in a single way) because 5^2 = 3^2 + 4^2, 10^2 = 6^2 + 8^2, 13^2 = 5^2 + 12^2, 17^2 = 8^2 + 15^2, and 20^2 = 12^2 + 16^2.
Since A046080(25) = 2 and A088111(3) = 2, we have that 25^2 can be written as a sum of two positive squares in two ways. Indeed, 25^2 = 7^2 + 24^2 = 15^2 + 20^2. Thus, a(3) = 25.
For 26 <= n <= 64, we see from sequence A046080 that n^2 cannot be written in more than 2 ways as a sum of two positive squares.
Since A046080(65) = 4, we see that 65^2 can be written as the sum of two positive squares in 4 ways. Indeed, 65^2 = 16^2 + 63^2 = 25^2 + 60^2 = 33^2 + 56^2 = 39^2 + 52^2. Thus, a(4) = 65.
(End)

R. M. Sternheimer, Additional Remarks Concerning The Pythagorean Triplets, Journal of Recreational Mathematics, Vol. 30, No. 1, pp. 45-48, 1999-2000, Baywood NY.

Ray Chandler, Table of n, a(n) for n = 1..307

Cf. A052199. Subsequence of A054994. Number of ways: see A088111. Where records occur in A046080.

from math import prod
from sympy import isprime
primes_congruent_1_mod_4 = [5]
def prime_4k_plus_1(i): # the i-th prime that is congruent to 1 mod 4
    while i>=len(primes_congruent_1_mod_4): # generate primes on demand
        n = primes_congruent_1_mod_4[-1]+4
        while not isprime(n): n += 4
        primes_congruent_1_mod_4.append(n)
    return primes_congruent_1_mod_4[i]
def generate_A054994():
    TO_DO = {(1,())}
    while True:
        radius, exponents = min(TO_DO)
        yield radius, exponents
        TO_DO.remove((radius, exponents))
        TO_DO.update(successors(radius,exponents))
def successors(r,exponents):
    for i,e in enumerate(exponents):
        if i==0 or exponents[i-1]>e:
            yield (r*prime_4k_plus_1(i), exponents[:i]+(e+1,)+exponents[i+1:])
    if exponents==() or exponents[-1]>0:
        yield (r*prime_4k_plus_1(len(exponents)), exponents+(1,))
n,record=0,-1
for radius,expo in generate_A054994():
    num_pyt = (prod((2*e+1) for e in expo)-1)//2
    if num_pyt>record:
        record = num_pyt
        n += 1
        print(radius, end="") # or record, for A088111
        if n==26: break # stop after 26 entries
        print(end=", ")
print() # Günter Rote, Sep 13 2023

Corrected and extended by Ray Chandler, Jan 12 2012

Name edited by Petros Hadjicostas, Jul 21 2019

A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

Original entry on oeis.org

1, 25, 325, 1105, 4225, 5525, 203125, 27625, 71825, 138125, 2640625, 160225, 17850625, 1221025, 1795625, 801125, 1650390625, 2082925, 49591064453125, 4005625, 44890625, 2158203125, 30525625, 5928325, 303460625, 53955078125, 35409725, 100140625

Offset: 1

Artur Jasinski, Nov 15 2006

nonn

The number must be strictly positive, but one of the squares may be zero, as we see from a(1) = 1 = 1^2 + 0^2 and a(2) = 25 = 3^2 + 4^2 = 5^0 + 0^2. - M. F. Hasler, Jul 07 2024

			a(3) = 325 is decomposable in 3 ways: 15^2 + 10^2 = 17^2 + 6^2 = 18^2 + 1^2.

Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).

Cf. A002144, A018782, A054994, A118882.

See A016032, A000446 and A093195 for other versions.

A124980(n)={for(a=1, oo, A000161(a)==n && return(a))} \\ R. J. Mathar, Nov 29 2006, edited by M. F. Hasler, Jul 07 2024

PD(n, L=n, D=Vecrev(divisors(n)[^1])) = { if(n>1, concat(vector(#D, i, if(D[i] > L, [], D[i] < n, [concat(D[i], P) | P <- PD(n/D[i], D[i])], [[n]]))), [[]])}
apply( {A124980(n)=vecmin([prod(i=1, #a, A002144(i)^(a[i]-1)) | a<-concat([PD(n*2,n), PD(n*2-1)])])}, [1..44]) \\ M. F. Hasler, Jul 07 2024

from sympy import divisors, isprime, prod
def PD(n, L=None): return [[]] if n==1 else [
    [d]+P for d in divisors(n)[:0:-1] if d <= (L or n) for P in PD(n//d, d)]
A2144=lambda upto=999: filter(isprime, range(5, upto, 4))
def A124980(n):
    return min(prod(a**(f-1) for a,f in zip(A2144(), P))
               for P in PD(n*2, n)+PD(n*2-1)) # M. F. Hasler, Jul 07 2024

a(n) = A000446(n), n > 1. - R. J. Mathar, Jun 15 2008
a(n) = min(A018782(2n-1), A018782(2n)).
a(n) = min { k > 0 | A000161(k) = n }. - M. F. Hasler, Jul 07 2024

More terms from R. J. Mathar, Nov 29 2006

Edited and extended by Ray Chandler, Jan 07 2012

A344473 Numbers of the form (q1^b1)(q2^b2)(q3^b3)(q4^b4)(q5^b5)... where q1=7, q2=13, q3=19, q4=31, q5=37, ... (A002476) and b1>=b2>=b3>=b4>=b5...

Original entry on oeis.org

1, 7, 49, 91, 343, 637, 1729, 2401, 4459, 8281, 12103, 16807, 31213, 53599, 57967, 84721, 117649, 157339, 218491, 375193, 405769, 593047, 753571, 823543, 1101373, 1529437, 1983163, 2626351, 2840383, 2989441, 4151329, 4877509, 5274997, 5764801, 7709611

Offset: 1

Jianing Song, May 20 2021

A343771 is a subsequence.

			12103 is a term since 12103 = 7^2 * 13 * 19.
22477 is not a term since 22477 = 7 * 13^2 * 19, the exponents are not nonincreasing.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A002476, A343771, A054994.

\\ following program for A054994
list_A344473(lim) =
{
    my(u = [1], v = List(), w = v, pr, t = 1);
    forprime(p = 7, ,
        if (p % 3 > 1, next);
        t *= p;
        if (t > lim,
            break);
        listput(w, t)
    );
    for (i = 1, #w,
        pr = 1;
        for (e = 1, logint(lim\ = 1, w[i]),
            pr *= w[i];
            for (j = 1, #u,
                t = pr * u[j];
                if (t > lim,
                    break);
                listput(v, t)
            )
        );
        if (w[i] ^ 2 < lim, u = Set(concat(Vec(v), u)); v = List());
    );
    Set(concat(Vec(v), u));
}
list_A344473(100000)

cp's OEIS Frontend

A097756 Table read by rows of A054994 ordered by A046080.

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A071383 Squared radii of the circles around (0,0) that contain record numbers of lattice points.

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A093195 Least number which is the sum of two distinct nonzero squares in exactly n ways.

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A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

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A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

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A000446 Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

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A052199 Numbers that are expressible as the sum of 2 distinct positive squares in more ways than any smaller number.

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A088959 Lowest numbers which are d-Pythagorean decomposable, i.e., square is expressible as sum of two positive squares in more ways than for any smaller number.

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