A025435 -id:A025435 - cp's OEIS frontend

A025441 Number of partitions of n into 2 distinct nonzero squares.

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

Offset: 0

David W. Wilson

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for sequences related to sums of squares

Cf. A060306 gives records; A052199 gives where records occur.
Cf. A000161, A000290, A010052, A025435, A157228, A053866, A145393, A093709.
Column k=2 of A341040.
Cf. A004439 (a(n)=0), A025302 (a(n)=1), A025303 (a(n)=2), A025304 (a(n)=3), A025305 (a(n)=4), A025306 (a(n)=5), A025307 (a(n)=6), A025308 (a(n)=7), A025309 (a(n)=8), A025310 (a(n)=9), A025311 (a(n)=10), A004431 (a(n)>0).

a025441 n = sum $ map (a010052 . (n -)) $
                      takeWhile (< n `div` 2) $ tail a000290_list
-- Reinhard Zumkeller, Dec 20 2013

Table[Count[PowersRepresentations[n, 2, 2], pr_ /; Unequal @@ pr && FreeQ[pr, 0]], {n, 0, 107}] (* Jean-François Alcover, Mar 01 2019 *)

a(n)=if(n>4,sum(k=1,sqrtint((n-1)\2),issquare(n-k^2)),0) \\ Charles R Greathouse IV, Jun 10 2016

a(n)=if(n<5,return(0)); my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)/2-issquare(n/2) \\ Charles R Greathouse IV, Jun 10 2016

from math import prod
from sympy import factorint
def A025441(n):
    f = factorint(n).items()
    return -int(not (any((e-1 if p == 2 else e)&1 for p,e in f) or n&1)) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 0 # Chai Wah Wu, Sep 08 2022

a(A025302(n)) = 1. - Reinhard Zumkeller, Dec 20 2013
a(n) = Sum_{ m: m^2|n } A157228(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019
a(n) = Sum_{i=1..floor((n-1)/2)} c(i) * c(n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020
a(n) = A000161(n) - A093709(n). - Andrey Zabolotskiy, Apr 12 2022

A001983 Numbers that are the sum of 2 distinct squares: of form x^2 + y^2 with 0 <= x < y.

Original entry on oeis.org

1, 4, 5, 9, 10, 13, 16, 17, 20, 25, 26, 29, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 73, 74, 80, 81, 82, 85, 89, 90, 97, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 130, 136, 137, 144, 145, 146, 148, 149, 153, 157, 160, 164

Offset: 1

N. J. A. Sloane

This sequence lists the values of A000404(n)/2 when A000404(n) is an even number. In other words, sequence lists integers n that are the average of two nonzero squares. - Altug Alkan, May 26 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A000404, subsequence of A001481, A004435 (complement), A025435, A004431.

Union of A000290 and A004431 excluding 0.

a001983 n = a001983_list !! (n-1)
a001983_list = [x | x <- [0..], a025435 x > 0]
-- Reinhard Zumkeller, Dec 20 2013

upto=200;max=Floor[Sqrt[upto]];s=Total/@((Subsets[Range[0,max], {2}])^2);Union[Select[s,#<=upto&]]  (* Harvey P. Dale, Apr 01 2011 *)
selQ[n_] := Select[ PowersRepresentations[n, 2, 2], 0 <= #[[1]] < #[[2]] &] != {}; Select[Range[200], selQ] (* Jean-François Alcover, Oct 03 2013 *)

list(lim)=my(v=List()); for(x=0,sqrtint(lim\4), for(y=x+1, sqrtint(lim\1-x^2), listput(v, x^2+y^2))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

A025435(a(n)) > 0. - Reinhard Zumkeller, Dec 20 2013

A004435 Positive integers that are not the sum of 2 distinct square integers.

Original entry on oeis.org

2, 3, 6, 7, 8, 11, 12, 14, 15, 18, 19, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 72, 75, 76, 77, 78, 79, 83, 84, 86, 87, 88, 91

Offset: 1

N. J. A. Sloane

nonn

A025435(a(n)) = 0. - Reinhard Zumkeller, Dec 20 2013

Cf. A001983 (complement).

a004435 n = a004435_list !! (n-1)
a004435_list = [x | x <- [1..], a025435 x == 0]
-- Reinhard Zumkeller, Dec 20 2013

Select[Range[100], Reduce[0 <= i < j && # == i^2 + j^2, {i, j}, Integers] === False &] (* Jean-François Alcover, Aug 01 2018 *)

A259285 Expansion of psi(x^2) * f(x, x^7) in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 23 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + x^2 + x^3 + x^6 + 2*x^7 + x^9 + x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^13 + q^29 + q^45 + q^61 + q^109 + 2*q^125 + q^157 + q^173 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000161, A025426, A025435, A025441, A134343, A259287.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^8] QPochhammer[ -x^2, x^8] QPochhammer[ -x^6, x^8] QPochhammer[ -x^7, x^8] QPochhammer[x^8]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 + x^(8 k - 1)) (1 + x^(8 k - 2)) (1 + x^(8 k - 6)) (1 + x^(8 k - 7)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];

{a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 13); for(u = (s+3)\-8, (s-3)\8, if( issquare( m - (8*u + 3)^2, &x) && (x%8==2 || x%8==6), c++))); c};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -1, 0, 0, 1, 0, -1, -1, 2, -1, -1, 0, 1, 0, 0, -1][k%16 + 1], 1 + x * O(x^n)), n))};

Number of solutions to 16*n + 13 = (8*u + 3)^2 + (8*v + 2)^2 where u,v in Z.
Euler transform of period 16 sequence [ 1, 0, 0, -1, 0, 1, 1, -2, 1, 1, 0, -1, 0, 0, 1, -2, ...].
a(9*n + 2) = A259287(n). a(9*n + 5) = a(9*n + 8) = 0.
-2 * a(n) = A134343(4*n + 3). a(n) = A000161(16*n + 13) = A025426(16*n + 13) = A025435(16*n + 13) = A025441(16*n + 13).

A259287 Expansion of psi(x^2) * f(x^3, x^5) in powers of x where psi(), f(, ) are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 23 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x^2 + x^3 + 2*x^5 + x^6 + x^7 + x^9 + x^11 + x^12 + x^14 + ...
G.f. = q^5 + q^37 + q^53 + 2*q^85 + q^101 + q^117 + q^149 + q^181 + q^197 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000161, A025426, A025435, A025441, A134343, A259285.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^8] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ -x^6, x^8] QPochhammer[x^8]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[(1 + x^(8 k - 2)) (1 + x^(8 k - 3)) (1 + x^(8 k - 5)) (1 + x^(8 k - 6)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];

{a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 5); for(u = (s+1)\-8, (s-1)\8, if( issquare( m - (8*u + 1)^2, &x) && (x%8==2 || x%8==6), c++))); c};

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, 0, -1, -1, 1, -1, 0, 0, 2, 0, 0, -1, 1, -1, -1, 0][k%16 + 1], 1 + x * O(x^n)), n))};

Number of solutions to 16*n + 5 = (8*u + 1)^2 + (8*v + 2)^2 where u,v in Z.
Euler transform of period 16 sequence [ 0, 1, 1, -1, 1, 0, 0, -2, 0, 0, 1, -1, 1, 1, 0, -2, ...].
a(9*n + 1) = a(9*n + 4) = 0. a(9*n + 7) = A259285(n).
-2 * a(n) = A134343(4*n + 1). a(n) = A000161(16*n + 5) = A025426(16*n + 5) = A025435(16*n + 5) = A025441(16*n + 5).

A317682 Number of partitions of n into a prime and two distinct squares.

Original entry on oeis.org

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

Offset: 0

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

As in A025435, zero is a valid square here.

			a(12)=4 counts 12 = 11 + 0^2 + 1^2 = 3 + 0^2 + 3^2 = 7 + 1^2 + 2^2 = 2 + 1^2 + 3^2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A025435, A317683 - A317685.

A317682 := proc(n)
    a := 0 ;
    p := 2;
    while p < n do
        a := a+A025435(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

A025435[n_] := Length[ PowersRepresentations[n, 2, 2]] - Boole[ IntegerQ[ Sqrt[2n]]];
a[n_] := Module[{s = 0, p}, For[p = 2, p <= n-1, p = NextPrime[p], s += A025435[n-p]]; s];
a /@ Range[0, 100] (* Jean-François Alcover, Apr 07 2020 *)

A317682(n,s=0)={forprime(p=2,n-1,s+=A025435(n-p));s} \\ M. F. Hasler, Aug 05 2018

a(n) = Sum_{primes p} A025435(n-p).

A347534 Number of partitions of n into at most 3 distinct squares.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Index entries for sequences related to sums of squares

Cf. A000290, A025435, A025442, A033461, A341040.

a[n_] := IntegerPartitions[n, 3, Select[Range[n], IntegerQ@Sqrt[#]&]] // Select[#, Union[#] == Sort[#]&]& // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 13 2021 *)

a(n) = Sum_{k=0..3} A341040(n,k). - Alois P. Heinz, Sep 08 2021

A369768 Numbers X such that X^2 + Y^2 = 3^(2*k) + 1 and X > Y > 0 and k is the ternary digit length of X-1.

Original entry on oeis.org

3, 9, 21, 27, 71, 81, 195, 233, 243, 711, 729, 1583, 1749, 2157, 2187, 6561, 14829, 15747, 19629, 19683, 57609, 59049, 141717, 154727, 175537, 177147, 385559, 394471, 414649, 422729, 446489, 462601, 468919, 482759, 488431, 504161, 515649, 524559, 529599, 530529

Offset: 1

A.H.M. Smeets, Jan 31 2024

The number of terms for a given k is A025435(3^(2*k)+1).

A.H.M. Smeets, Table of n, a(n) for n = 1..191

Cf. A004018, A025435.

Cf. A369703 (base 2), this sequence (base 3), A369769 (base 5), A368418 (base 10).

A369769 Numbers X such that X^2 + Y^2 = 5^(2*k) + 1 and X > Y > 0 and k is the quintal digit length of X-1.

Original entry on oeis.org

5, 25, 115, 125, 551, 625, 2551, 2885, 3049, 3125, 15575, 15625, 72115, 78125, 303551, 390625, 1461799, 1790635, 1802885, 1847551, 1857449, 1946801, 1952875, 1953125, 8053505, 8468945, 9734425, 9765625, 36631645, 43890449, 45072115, 48828125, 215049449, 215418199

Offset: 1

A.H.M. Smeets, Jan 31 2024

The number of terms for a given k is A025435(5^(2*k)+1).

A.H.M. Smeets, Table of n, a(n) for n = 1..116

Cf. A025435.

Cf. A369703 (base 2), A369768 (base 3), this sequence (base 5), A368418 (base 10).

A357069 Number of partitions of n into at most 4 distinct positive squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 0, 2, 2, 0, 0, 2, 3, 1, 1, 2, 2, 0, 1, 1, 1, 1, 0, 2, 3, 1, 1, 4, 2, 0, 1, 2, 2, 1, 0, 1, 4, 2, 0, 2, 4, 1, 1, 3, 1, 1, 2, 3, 3, 1, 0, 3, 5, 2, 0, 2, 4, 2, 0, 1, 3, 2, 2, 4

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Index entries for sequences related to sums of squares

Cf. A000290, A002635, A025435, A025443, A033461, A341040, A347534, A347586, A347711.

a(n) = Sum_{k=0..4} A341040(n,k). - Alois P. Heinz, Oct 25 2022

cp's OEIS Frontend

A025441 Number of partitions of n into 2 distinct nonzero squares.

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A001983 Numbers that are the sum of 2 distinct squares: of form x^2 + y^2 with 0 <= x < y.

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A004435 Positive integers that are not the sum of 2 distinct square integers.

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A259285 Expansion of psi(x^2) * f(x, x^7) in powers of x where psi(), f(,) are Ramanujan theta functions.

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A259287 Expansion of psi(x^2) * f(x^3, x^5) in powers of x where psi(), f(, ) are Ramanujan theta functions.

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A317682 Number of partitions of n into a prime and two distinct squares.

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A347534 Number of partitions of n into at most 3 distinct squares.

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