A000161 -id:A000161 - cp's OEIS frontend

A217462 a(n) is the sum of total number of nonnegative integer solutions to each of a^2 + b^2 = n, a^2 + 2b^2 = n, a^2 + 3b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).

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

Offset: 1

V. Raman, Oct 04 2012

nonn

Note: For the equation a^2 + b^2 = n, if there are two solutions (a,b) and (b,a), then they will be counted only once.
The sequences A216501 and A216671 give how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to.
1, 2, 3, 7 are the first four numbers, with the class number 1.
"If a composite number C is of the form a^2 + kb^2 for some integers a & b, then every prime factor of C raised to an odd power is of the form c^2 + kd^2 for some integers c & d."
This statement is only true for k = 1, 2, 3.
For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.
A number can be written as a^2 + b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
A number can be written as a^2 + 2b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
A number can be written as a^2 + 3b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
A number can be written as a^2 + 7b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power, and the exponent of 2 is not 1.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Cf. A216501, A216671.
Cf. A217868 (related sequence of this when the order does matter for the equation a^2 + b^2 = n).
Cf. A216501 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a > 0, b > 0).
Cf. A216671 (how many of the four k values, k = 1, 2, 3, 7 does the equation a^2 + k*b^2 = n have a solution to, with a >= 0, b >= 0).
Cf. A000161 (number of solutions to n = a^2+b^2 (when the solutions (a, b) and (b, a) are being counted as the same) with a >= 0, b >= 0).
Cf. A216282 (number of solutions to n = a^2+2*b^2 with a >= 0, b >= 0).
Cf. A119395 (number of solutions to n = a^2+3*b^2 with a >= 0, b >= 0).
Cf. A216512 (number of solutions to n = a^2+7*b^2 with a >= 0, b >= 0).

for(n=1,100,sol=0;for(x=0,100,if(issquare(n-x*x)&&n-x*x>=0&&x*x<=n-x*x,sol++);if(issquare(n-2*x*x)&&n-2*x*x>=0,sol++);if(issquare(n-3*x*x)&&n-3*x*x>=0,sol++);if(issquare(n-7*x*x)&&n-7*x*x>=0,sol++));printf(sol","))

A306358 Odd numbers which are the sum of two squares in two or more different ways.

Original entry on oeis.org

25, 65, 85, 125, 145, 169, 185, 205, 221, 225, 265, 289, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 585, 625, 629, 685, 689, 697, 725, 745, 765, 785, 793, 841, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1073, 1105, 1125, 1145, 1157, 1165, 1189, 1205, 1225, 1241

Offset: 1

Joerg Arndt, Feb 10 2019

nonn

Odd terms of A118882.

Odd numbers k such that A000161(k) >= 2.

			The decompositions of the first terms are
25: [[4, 3], [5, 0]]
65: [[7, 4], [8, 1]]
85: [[7, 6], [9, 2]]
125: [[10, 5], [11, 2]]
145: [[9, 8], [12, 1]]
169: [[12, 5], [13, 0]]
185: [[11, 8], [13, 4]]
205: [[13, 6], [14, 3]]
221: [[11, 10], [14, 5]]
225: [[12, 9], [15, 0]]
265: [[12, 11], [16, 3]]
289: [[15, 8], [17, 0]]
305: [[16, 7], [17, 4]]
325: [[15, 10], [17, 6], [18, 1]]
365: [[14, 13], [19, 2]]
377: [[16, 11], [19, 4]]

A000161(n)=sum(k=sqrtint((n-1)\2)+1, sqrtint(n), issquare(n-k^2));
is(n)=if(n%2==1, A000161(n)>1, 0);
select(is,vector(1300,n,n))

from itertools import count, islice
from math import prod
from sympy import factorint
def A306358_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue+1-(startvalue&1),1),2):
        f = factorint(n)
        if 1>1):
            yield n
A306358_list = list(islice(A306358_gen(),30)) # Chai Wah Wu, Sep 09 2022

A328804 The maximum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 4, 5, 4, 5, 6, 6, 7, 6, 7, 8, 8, 6, 7, 8, 9, 9, 7, 10, 10, 9, 10, 11, 8, 11, 10, 12, 11, 12, 12, 9, 10, 13, 13, 12, 13, 14, 14, 11, 12, 14, 13, 15, 14, 15, 11, 12, 15, 16, 16, 16, 15, 12, 17, 16, 14, 17, 15, 17, 16, 18, 18, 17, 18, 15, 16

Offset: 1

Peter Kagey, Oct 27 2019

nonn

			For n = 14, A001481(14) = 25 = 0^2 + 5^2 = 3^2 + 4^2, so a(14) = max{0+5, 3+4} = 7.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000161, A001481, A328803.

from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
from sympy import factorint
def A328804_gen(): # generator of terms
    return map(lambda n: max((a+b for a, b in diop_DN(-1,n))), filter(lambda n:(lambda m:all(d&3!=3 or m[d]&1==0 for d in m))(factorint(n)), count(0)))
A328804_list = list(islice(A328804_gen(),30)) # Chai Wah Wu, Sep 09 2022

A379925 Numbers k for which nonnegative integers x and y exist such that x^2 + y^2 = k and x + y is a square.

Original entry on oeis.org

0, 1, 8, 10, 16, 41, 45, 53, 65, 81, 128, 130, 136, 146, 160, 178, 200, 226, 256, 313, 317, 325, 337, 353, 373, 397, 425, 457, 493, 533, 577, 625, 648, 650, 656, 666, 680, 698, 720, 746, 776, 810, 848, 890, 936, 986, 1040, 1098, 1160, 1201, 1205, 1213, 1225, 1226

Offset: 1

Felix Huber, Jan 25 2025

Numbers k for which exists at least one solution to k = x^2 + (z^2 - x)^2 in integers x and z with x >= 0 and z >= sqrt(2*x).

Subsequence of A001481.

			10 is in the sequence because 10 = 1^2 + 3^2 and 1 + 3 = 2^2.
81 is in the sequence because 81 = 0^2 + 9^2 and 0 + 9 = 3^2.

Felix Huber, Table of n, a(n) for n = 1..10000
Wikipedia, Sum of two squares theorem

Cf. A000161, A000290, A001481, A004018, A025284-A025320, A091072, A125110, A118882, A125022, A380072, A380073, A380074.

# Calculates the first 10005 terms.
A379925:=proc(K)
    local i,j,L;
    L:={};
    for i from 0 to floor(sqrt((K+1)^2)/2) do
        for j from 0 to floor(sqrt((K+1)^2/2-i^2)) do
            if issqr(i+j) then
                L:=L union {i^2+j^2}
            fi
        od
    od;
    return op(L)
end proc;
A379925(1737);

isok(n)=my(x=0, r=0); while(x<=sqrt(n) && r==0, if(issquare(n-x^2) && issquare(x+sqrtint(n-x^2)), r=1); x++); r; \\ Michel Marcus, Feb 10 2025

k = m^(4*j) is in the sequence for nonnegative integers m and j (not both 0) because x = 0 and z = m^j is a solution to m^(4*j) = x^2 + (z^2 - x)^2.

A116905 Number of partitions of n-th 3-almost prime into 2 squares.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Jonathan Vos Post, Mar 15 2006

See also A000161 Number of partitions of n into 2 squares (when order does not matter and zero is allowed).

			a(1) = 1 because A014612(1) = 8 = 2^2 + 2^2, the unique sum of squares.
a(2) = 0 because A014612(2) = 12 has no decomposition into sum of 2 squares because it has a prime factor p == 3 (mod 4) with an odd power.
a(11) = 2 because A014612(11) = 50 = 2*5^2 = 1^2 + 7^2 = 5^2 + 5^2.
a(30) = 2 because A014612(30) = 125 = 5^3 = 2^2 + 11^2 = 5^2 + 1^0.
a(31) = 2 because A014612(31) = 130 = 2*5*13 = 3^2 + 11^2 = 7^2 + 9^2.
a(39) = 2 because A014612(39) = 170 = 2*5*17 = 1^2 + 13^2 = 7^2 + 11^2.

Cf. A000161, A014612.

a(n) = A000161(A014612(n)).

cp's OEIS Frontend

A217462 a(n) is the sum of total number of nonnegative integer solutions to each of a^2 + b^2 = n, a^2 + 2*b^2 = n, a^2 + 3*b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).

Views

Author

Keywords

Comments

References

Crossrefs

Programs

PARI

A306358 Odd numbers which are the sum of two squares in two or more different ways.

Views

Author

Keywords

Comments

Examples

Programs

PARI

Python

A328804 The maximum value of j + k where j and k are positive integers with j^2 + k^2 = A001481(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

A379925 Numbers k for which nonnegative integers x and y exist such that x^2 + y^2 = k and x + y is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A116905 Number of partitions of n-th 3-almost prime into 2 squares.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A217462 a(n) is the sum of total number of nonnegative integer solutions to each of a^2 + b^2 = n, a^2 + 2b^2 = n, a^2 + 3b^2 = n and a^2 + 7*b^2 = n. (Order does not matter for the equation a^2+b^2 = n).