A025426 -id:A025426 - cp's OEIS frontend

A025286 Numbers that are the sum of 2 nonzero squares in exactly 3 ways.

325, 425, 650, 725, 845, 850, 925, 1025, 1250, 1300, 1325, 1445, 1450, 1525, 1690, 1700, 1825, 1850, 2050, 2225, 2425, 2525, 2600, 2650, 2725, 2825, 2873, 2890, 2900, 2925, 3050, 3125, 3380, 3400, 3425, 3650, 3700, 3725, 3757, 3825, 3925, 4100, 4205

Offset: 1

David W. Wilson

nonn

A025426(a(n)) = 3. - Reinhard Zumkeller, Feb 26 2015

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

Cf. A025426.

a025286 n = a025286_list !! (n-1)
a025286_list = filter ((== 3) . a025426) [1..]
-- Reinhard Zumkeller, Feb 26 2015

A025294 Numbers that are the sum of 2 nonzero squares in 3 or more ways.

Original entry on oeis.org

325, 425, 650, 725, 845, 850, 925, 1025, 1105, 1250, 1300, 1325, 1445, 1450, 1525, 1625, 1690, 1700, 1825, 1850, 1885, 2050, 2125, 2210, 2225, 2405, 2425, 2465, 2525, 2600, 2650, 2665, 2725, 2825, 2873, 2890, 2900, 2925, 3050, 3125, 3145, 3250, 3380

Offset: 1

David W. Wilson

nonn

A025426(a(n)) > 2. - Reinhard Zumkeller, Feb 26 2015

Robert Israel and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 2508 terms from Robert Israel)
Index entries for sequences related to sums of squares

Complement of A025285 relative to A007692. - Washington Bomfim, Oct 24 2010

Cf. A025426.

a025294 n = a025294_list !! (n-1)
a025294_list = filter ((> 2) . a025426) [1..]
-- Reinhard Zumkeller, Feb 26 2015

  N:= 100000: # generate all entries <=N
SSQ:= {}: SSQ2:= {}: SSQ3:= {}:
for a from 1 to floor(sqrt(N)) do
  for b from a to floor(sqrt(N-a^2)) do
    n:= a^2 + b^2;
    if member(n,SSQ2) then SSQ3:= SSQ3 union {n}
    elif member(n,SSQ) then SSQ2:= SSQ2 union {n}
    else SSQ:= SSQ union {n}
    end if
end do end do:
SSQ3;  # Robert Israel, Jan 20 2013

okQ[n_] := Length[Select[PowersRepresentations[n, 2, 2][[All, 1]], Positive] ] > 2; Select[Range[5000], okQ] (* Jean-François Alcover, Mar 04 2019 *)

A273238 Least number k such that k^3 is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

2, 5, 25, 50, 125, 625, 1250, 65, 15625, 31250, 78125, 390625, 781250, 325, 9765625, 19531250, 48828125, 244140625, 488281250, 1625, 6103515625, 12207031250, 30517578125, 4225, 8450, 8125, 3814697265625, 7629394531250, 19073486328125, 95367431640625

Offset: 1

Altug Alkan, May 18 2016

nonn

			a(1) = 2 because 2^3 = 2^2 + 2^2.
a(2) = 5 because 5^3 = 5^2 + 10^2 = 2^2 + 11^2.
a(3) = 25 because 25^3 = 35^2 + 120^2 = 44^2 + 117^2 = 75^2 + 100^2.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A006339, A016032, A025426, A084888.

Function[t, FirstPosition[t, #] & /@ Range@ 8]@ Map[Length@ Select[ PowersRepresentations[#^3, 2, 2], ! MemberQ[#, 0] &] &, Range[2 10^3]] // Flatten (* Michael De Vlieger, May 18 2016 *)
(* code for first 100 terms *) nR[n_] := (SquaresR[2, n] + Plus @@ Pick[{-4, 4}, IntegerQ /@ Sqrt[{n, n/2}]])/8; c[w_] := Floor[1/2 Times @@ (3 w + 1)]; q[1] = 2; q[n_] := Min[Reap[Do[ x = Times @@ (Take[{5, 13, 17, 29}, Length[e]]^e); If[c[e] == n && nR[x^3] == n, Sow[x]]; If[c[e] + 1 == n && nR[8 x^3] == n, Sow[2 x]], {e, Join[Transpose[{ Range@ 80}], Join @@ (IntegerPartitions[#, 4] & /@ Range[21]) ]}]][[2, 1]]]; Array[q, 100] (* Giovanni Resta, May 18 2016 *)

A025426(n)=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
a(n)=my(k=1); while(A025426(k++^3)!=n, ); k
first(n)=my(v=vector(n),t,k); while(1, t=A025426(k++^3); if(t>0 && t<=n && v[t]==0, v[t]=k; if(factorback(v), return(v)))) \\ Charles R Greathouse IV, May 18 2016

a(10)-a(30) from Giovanni Resta, May 18 2016

A321435 Expansion of Product_{1 <= i <= j} (1 + x^(i^2 + j^2)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 1, 2, 0, 3, 0, 2, 3, 1, 4, 1, 2, 4, 1, 6, 3, 4, 6, 2, 7, 5, 6, 8, 5, 9, 7, 9, 10, 9, 12, 10, 13, 14, 13, 18, 13, 19, 17, 18, 25, 19, 28, 24, 25, 33, 26, 36, 35, 33, 46, 35, 47, 48, 44, 61, 48, 62, 65, 60, 78, 68, 79, 87, 79, 101, 93

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A025426, A321423, A321428, A321436, A321437.

N:= 100: # for a(0)..a(N)
P:= 1:
for i from 1 to floor(sqrt(N)) do
  for j from i while i^2 + j^2 <= N do
    P:= P * (1 + x^(i^2 + j^2))
od od:
S:= series(P,x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Apr 21 2024

G.f.: Product_{k>0} (1 + x^k)^A025426(k).

A321436 Expansion of Product_{1 <= i <= j} (1 - x^(i^2 + j^2)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Nov 09 2018

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A025426, A321430, A321435, A321437.

N:= 100: # for a(1)..a(N)
P:= 1:
for i from 1 to floor(sqrt(N)) do
  for j from i while i^2 + j^2 <= N do
    P:= P * (1 - x^(i^2 + j^2))
od od:
S:= series(P,x,N+1):
seq(coeff(S,x,k),k=0..N): # Robert Israel, Apr 21 2024

G.f.: Product_{k>0} (1 - x^k)^A025426(k).

A007511 a(n) is the smallest number greater than a(n-1) that is expressible as the sum of two squares in more ways than a(n-1).

Original entry on oeis.org

2, 50, 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

Gabriel Cunningham (gcasey(AT)mit.edu), Feb 29 2004

nonn

Sequence provides the locations of records in A025426 (nonzero squares), rather than in A000161 (definition of squares includes zeros). - R. J. Mathar, Jun 06 2007

J. Meeus, Note

Cf. A048610.

a(12)-a(18) from Donovan Johnson, Sep 03 2008
a(19)-a(24) from Donovan Johnson, Jul 01 2009
a(25)-a(26) from Donovan Johnson, Aug 30 2011

A216284 Number of solutions to the equation x^4+y^4 = n with x >= y > 0.

Original entry on oeis.org

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

V. Raman, Sep 03 2012

nonn

			From _Antti Karttunen_, Aug 28 2017: (Start)
For n = 2 there is one solution: 2 = 1^4 + 1^4, thus a(2) = 1.
For n = 17 there is one solution: 17 = 2^4 + 1^4, thus a(17) = 1.
For n = 635318657 we have two solutions: 635318657 = 158^4 + 59^4 = 134^4 + 133^4, thus a(635318657) = 2. Note that this is the first point where the sequence attains value greater than 1. See _Charles R Greathouse IV_'s Jan 12 2017 comment in A216280.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A025455, A025446, A000161, A025426, A216280.

(define (A216284 n) (let loop ((x (A255270 n)) (s 0)) (let* ((x4 (A000583 x)) (y4 (- n x4))) (if (< x4 y4) s (loop (- x 1) (+ s (if (and (> y4 0) (= (A000583 (A255270 y4)) y4)) 1 0))))))) ;; Antti Karttunen, Aug 28 2017

a(n) <= A216280(n). - Antti Karttunen, Aug 28 2017

Definition edited to match the given data and the second part of offset (635318657) explicitly added by Antti Karttunen, Aug 28 2017

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).

A317685 Number of partitions of n into a prime and two positive squares.

Original entry on oeis.org

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

Offset: 0

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

As in A025426, the two squares do not need to be distinct.

			a(7) = 2 counts 7 = 5 + 1^2 + 1^2 = 2 + 1^2 + 2^2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A025426, A317682 - A317684.

A317685 := proc(n)
    a := 0 ;
    p := 2;
    while p <= n do
        a := a+A025426(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

p2sQ[{a_,b_,c_}]:=PrimeQ[a]&&AllTrue[{Sqrt[b],Sqrt[c]},IntegerQ]||PrimeQ[b] && AllTrue[{Sqrt[c],Sqrt[a]},IntegerQ]||PrimeQ[c]&&AllTrue[{Sqrt[b],Sqrt[a]},IntegerQ]; Table[Count[IntegerPartitions[n,{3}],?(p2sQ[#]&)],{n,0,80}] (* _Harvey P. Dale, Mar 10 2023 *)

a(n) = Sum_{primes p} A025426(n-p).

cp's OEIS Frontend

A025286 Numbers that are the sum of 2 nonzero squares in exactly 3 ways.

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A025294 Numbers that are the sum of 2 nonzero squares in 3 or more ways.

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Maple

Mathematica

A273238 Least number k such that k^3 is the sum of two nonzero squares in exactly n ways.

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Extensions

A321435 Expansion of Product_{1 <= i <= j} (1 + x^(i^2 + j^2)).

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Maple

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A321436 Expansion of Product_{1 <= i <= j} (1 - x^(i^2 + j^2)).

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Maple

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A007511 a(n) is the smallest number greater than a(n-1) that is expressible as the sum of two squares in more ways than a(n-1).

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A216284 Number of solutions to the equation x^4+y^4 = n with x >= y > 0.

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Scheme

Formula

Extensions

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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