A000161 -id:A000161 - cp's OEIS frontend

A317684 Number of partitions of n into a prime and two squares.

0, 0, 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 4, 2, 5, 3, 3, 4, 5, 5, 6, 4, 6, 4, 4, 2, 7, 6, 5, 5, 7, 6, 6, 4, 4, 7, 7, 5, 10, 4, 6, 8, 8, 6, 8, 5, 9, 9, 7, 4, 8, 8, 8, 9, 10, 8, 10, 6, 6, 9, 9, 6, 14, 6, 6, 10, 10, 10, 12, 8, 10, 12, 9, 6, 12, 10, 11, 11, 12, 7

Offset: 0

R. J. Mathar, Michel Marcus, Aug 04 2018

As in A000161, the squares may be zero and do not need to be distinct.

			a(11) = 4 counts 11 = 11+0^2+0^2 = 7+0^2+2^2 = 2+0^2+3^2 = 3+2^2+2^2.

Robert Israel, Table of n, a(n) for n = 0..10000
C. Hooley, On the representation of a number as the sum of two squares and a prime, Acta Mathem. 97 (1957) 189-210

Cf. A000161, A317682-A317685.

A317684 := proc(n)
    a := 0 ;
    p := 2;
    while p <= n do
        a := a+A000161(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

a(n) = Sum_{primes p} A000161(n-p).

A356208 a(n) is the number of occurrences of n in A133388.

Original entry on oeis.org

2, 3, 4, 4, 5, 7, 6, 8, 8, 9, 9, 10, 10, 12, 13, 12, 12, 15, 14, 17, 16, 16, 17, 18, 19, 18, 19, 20, 18, 24, 20, 22, 25, 22, 27, 26, 23, 25, 25, 29, 26, 30, 27, 31, 32, 32, 24, 33, 33, 34, 32, 32, 35, 37, 36, 37, 38, 32, 35, 44, 36, 41, 41, 40, 42, 45, 39, 43, 42

Offset: 1

Hugo Pfoertner, Sep 07 2022

nonn

Cf. A000161, A001481, A133388, A356209.

from sympy.solvers.diophantine.diophantine import diop_DN
def A356208(n): return sum(1 for m in range(1,(n**2<<1)+1) if n==max((a for a, b in diop_DN(-1,m)),default=0)) # Chai Wah Wu, Sep 08 2022

A356209 a(n) is the position of the latest occurrence of n in A133388.

Original entry on oeis.org

2, 8, 18, 32, 41, 72, 98, 128, 162, 181, 242, 288, 313, 392, 421, 512, 514, 648, 722, 761, 882, 968, 1058, 1152, 1201, 1301, 1458, 1568, 1466, 1741, 1922, 2048, 2178, 2056, 2381, 2592, 2594, 2888, 2817, 3121, 3202, 3528, 3698, 3872, 3789, 4232, 4418, 4608, 4802, 4804, 5101

Offset: 1

Hugo Pfoertner, Sep 07 2022

nonn

Cf. A000161, A001481, A009003, A356208.

from sympy.solvers.diophantine.diophantine import diop_DN
def A356209(n):
    for m in range(n**2<<1,0,-1):
        if n==max((a for a, b in diop_DN(-1,m)),default=0):
            return m # Chai Wah Wu, Sep 08 2022

a(k) = 2*k^2 for k not in A009003.

A073092 Number of numbers of the form x^2 + y^2 (0 <= x <= y) less than or equal to n.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 13, 13, 13, 13, 13, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 27, 27, 28, 29, 29, 29, 29, 29, 30, 30, 30, 31, 31, 31, 32, 34, 34, 34, 35, 35, 35, 35, 36, 37, 38

Offset: 0

Benoit Cloitre, Aug 18 2002

			0^2 + 0^2, 0^2 + 1^2, 1^2 + 1^2 are less than or equal to 2 hence a(2) = 3.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000161, A001481.

Cf. A057655.

Accumulate @ Table[Length @ PowersRepresentations[n, 2, 2], {n, 0, 100}] (* Amiram Eldar, Mar 08 2020 *)

a(n)=sum(x=0,n,sum(y=0,x,if((sign(x^2+y^2-n)+1)*sign(x^2+y^2-n),0,1)))

from itertools import count, islice
from math import prod
from sympy import factorint
def A073092_gen(): # generator of terms
    yield (c:=1)
    for n in count(1):
        f = factorint(n)
        c += int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1)
        yield c
A073092_list = list(islice(A073092_gen(),30)) # Chai Wah Wu, Sep 08 2022

a(n) = Sum_{k=0..n} A000161(k).

a(n) is asymptotic to Pi*n/8.

A085625 Numbers that are the sum of 2 squares in exactly 2 ways.

Original entry on oeis.org

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 338, 340, 365, 370, 377, 400, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 629, 676, 680, 685, 689, 697, 730

Offset: 1

Hugo Pfoertner, Jul 09 2003

nonn

Wells erroneously writes that this sequence begins as 50, 65, 85, 145, ... . - Stefano Spezia, Sep 07 2024

			a(3) = 65 because 65 = 8^2 + 1^2 = 7^2 + 4^2;
a(4) = 85 because 85 = 9^2 + 2^2 = 7^2 + 6^2.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.

Robert Price, Table of n, a(n) for n = 1..16794

Cf. A000161, A000446, A224443, A294282, A295153, A295489, A295748.

Select[Range[730], Length[PowersRepresentations[#,2,2]]==2 &] (* Stefano Spezia, Sep 07 2024 *)

n such that A000161(n) = 2.

A088918 Number of representations of n as sum of two squares of distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 23 2003

nonn

a(n) <= A000161(n); a(A088909(n)) > 0;

a(A088919(n)) = n and a(k) <> n for k < A088919(n).

			a(410)=2, see A088919.

Index entries for sequences related to sums of squares

Cf. A002654.

A116852 Number of partitions of n-th semiprime into 2 squares.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 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).
From Robert Israel, Jun 10 2020: (Start)
  a(1)=1 if A001358(n) = p^2 where p is not in A002144.
  a(n)=1 if A001358(n) = 2*p where p is in A002144.
  a(n)=2 if A001358(n) = p*q where p and q are in A002144 (not necessarily distinct).
  a(n)=0 otherwise. (End)

			a(1) = 1 because semiprime(1) = 4 = 0^2 + 2^2, the unique sum of squares.
a(2) = 0 because semiprime(2) = 6 has no decomposition into sum of 2 squares because it has a prime factor p == 3 (mod 4) with an odd power.
a(3) = 1 because semiprime(3) = 9 = 0^2 + 3^2, the unique sum of squares.
a(4) = 1 because semiprime(4) = 10 = 2*5 = 1^2 + 3^2.
a(9) = 2 because semiprime(9) = 25 = 0^2 + 5^2 = 3^2 + 4^2, two distinct ways.
a(23) = 2 because semiprime(23) = 65 = 5*13 = 1^2 + 8^2 = 4^2 + 7^2.
a(28) = 2 because semiprime(28) = 85 = 5*17 = 2^2 + 9^2 = 6^2 + 7^2.
a(49) = 2 because semiprime(49) = 145 = 5*29 = 1^2 + 12^2 = 8^2 + 9^2.
a(56) = 2 because semiprime(56) = 169 = 0^2 + 13^2 = 5^2 + 12^2.
a(60) = 2 because semiprime(60) = 185 = 5*37 = 4^2 + 13^2 = 8^2 + 11^2.

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

Cf. A000161, A001358.

R:= NULL: count:= 0:
for n from 4 while count < 100 do
  if numtheory:-bigomega(n) = 2 then
    count:= count+1;
    F:= ifactors(n)[2];
    if nops(F) = 1 then
      if F[1][1] mod 4 = 1 then v:= 2
      else v:= 1
      fi
    elif F[1][1]=2 and F[2][1] mod 4 = 1 then v:= 1
    elif F[1][1] mod 4 = 1 and F[2][1] mod 4 = 1 then v:= 2
    else v:= 0
    fi;
    R:= R, v;
  fi
od:
R; # Robert Israel, Jun 10 2020

a(n) = A000161(A001358(n)).

More terms from Giovanni Resta, Jun 15 2016

A171099 a(n) = number of solutions (x,y) (with 0 <= x <= y) to x(x+1)/2 + y(y+1)/2 = n!.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 2, 0, 0, 1, 0, 2, 1, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 8, 1, 2, 0, 0, 4, 4, 16, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 1, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 2, 0, 16, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 16, 2, 4, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Sep 24 2010, based on a posting by R. K. Guy to the Sequence Fans Mailing List, Sep 10 2010

nonn

			Initial solutions: (x,y,n) = (0,1,0), (0,1,1), (1,1,2), (0,3,3), (2,2,3), (2,6,4), (0,15,5), (5,14,5), (45,89,7), (89,269,8), (210,825,9), (760,2610,10), (1770,2030,10), none for n = 11 or 12, one for n = 13 (71504,85680,13) (found by _Ed Pegg Jr_), etc.

Cf. A000161, A152089 (n for which no solutions exist), A180590 (n for which solutions exist).

a(n) = A000161(8*n! + 2). - Max Alekseyev, Dec 12 2011

Corrected and extended (with data from Georgi Guninski, at the suggestion of N. J. A. Sloane) by D. S. McNeil, Sep 26 2010

A173256 Partial sums of A001481.

Original entry on oeis.org

0, 1, 3, 7, 12, 20, 29, 39, 52, 68, 85, 103, 123, 148, 174, 203, 235, 269, 305, 342, 382, 423, 468, 517, 567, 619, 672, 730, 791, 855, 920, 988, 1060, 1133, 1207, 1287, 1368, 1450, 1535, 1624, 1714, 1811, 1909, 2009, 2110, 2214, 2320, 2429, 2542, 2658, 2775

Offset: 1

Jonathan Vos Post, Feb 14 2010

nonn

The subsequence of primes in this sequence begins 3, 7, 29, 103, 269, 619, 1811, 3271.

			a(66) = 0 + 1 + 2 + 4 + 5 + 8 + 9 + 10 + 13 + 16 + 17 + 18 + 20 + 25 + 26 + 29 + 32 + 34 + 36 + 37 + 40 + 41 + 45 + 49 + 50 + 52 + 53 + 58 + 61 + 64 + 65 + 68 + 72 + 73 + 74 + 80 + 81 + 82 + 85 + 89 + 90 + 97 + 98 + 100 + 101 + 104 + 106 + 109 + 113 + 116 + 117 + 121 + 122 + 125 + 128 + 130 + 136 + 137 + 144 + 145 + 146 + 148 + 149 + 153 + 157 + 160 = 4876.

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

Cf. A001481, A022544, A004018, A000161, A002654, A064533, A000404, A002828, A000378, A025284-A025320, A125110, A091072.

N:= 1000:
A001481:= sort(convert({seq(seq(x^2+y^2, y=0..floor(sqrt(N-x^2))),x=0..floor(sqrt(N)))},list)):
ListTools:-PartialSums(A001481); # Robert Israel, Mar 15 2016

from itertools import count, accumulate, islice
from sympy import factorint
def A173256_gen(): # generator of terms
    return accumulate(filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(0)))
A173256_list = list(islice(A173256_gen(),30)) # Chai Wah Wu, Jun 27 2022

a(n) = Sum_{i=1..n} A001481(i) = Sum_{i=1..n} (numbers that are the sum of 2 nonnegative squares) = Sum_{i=1..n} (numbers n such that i = x^2 + y^2 has a solution in nonnegative integers x, y).

a(21) corrected by Robert Israel, Mar 15 2016

A181570 Primes in A050798.

Original entry on oeis.org

7, 13, 17, 23, 31, 37, 41, 53, 67, 89, 97, 103, 109, 113, 127, 137, 149, 151, 163, 167, 179, 197, 211, 223, 227, 229, 241, 263, 269, 277, 281, 283, 311, 331, 347, 359, 367, 373, 383, 389, 397, 419, 431, 433, 439, 479, 491, 503, 509, 541, 547, 587, 601, 617, 619, 653, 673, 677, 683, 691, 709

Offset: 1

Jonathan Vos Post, Jan 29 2011

Primes p such that p^2 + 1 is expressible as the sum of two nonzero squares in exactly two ways.

Cf. A000040, A000161, A050798.

A050798 INTERSECTION A000040. {p in A000040 such that A000161(p^2 + 1) = 2}.

cp's OEIS Frontend

A317684 Number of partitions of n into a prime and two squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A356208 a(n) is the number of occurrences of n in A133388.

Views

Author

Keywords

Crossrefs

Programs

Python

A356209 a(n) is the position of the latest occurrence of n in A133388.

Views

Author

Keywords

Crossrefs

Programs

Python

Formula

A073092 Number of numbers of the form x^2 + y^2 (0 <= x <= y) less than or equal to n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A085625 Numbers that are the sum of 2 squares in exactly 2 ways.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A088918 Number of representations of n as sum of two squares of distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A116852 Number of partitions of n-th semiprime into 2 squares.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A171099 a(n) = number of solutions (x,y) (with 0 <= x <= y) to x*(x+1)/2 + y*(y+1)/2 = n!.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A173256 Partial sums of A001481.

A171099 a(n) = number of solutions (x,y) (with 0 <= x <= y) to x(x+1)/2 + y(y+1)/2 = n!.