A000161 -id:A000161 - cp's OEIS frontend

A133105 Number of partitions of n^4 into n distinct nonzero squares.

1, 0, 1, 0, 21, 266, 2843, 55932, 884756, 13816633, 283194588, 5375499165, 125889124371, 3202887665805, 80542392920980, 2270543992935431, 64253268814048352, 1892633465941308859, 59116753827795287519, 1886846993941912938452

Offset: 1

Hugo Pfoertner, Sep 12 2007

nonn

			a(3)=1 because there is exactly one way to express 3^4 as the sum of 3 distinct nonzero squares: 81 = 1^2 + 4^2 + 8^2.

Robert Gerbicz, May 09 2008, Table of n, a(n) for n = 1..20

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.

Cf. A133104 (number of ways to express n^4 as a sum of n nonzero squares), A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares).

a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n

a(10) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

a(11) onwards from Robert Gerbicz, May 09 2008

A143575 Numbers m such that A143574(m) = m.

Original entry on oeis.org

0, 1, 4, 5, 9, 10, 13, 16, 17, 20, 26, 29, 34, 36, 37, 40, 41, 45, 49, 52, 53, 58, 61, 64, 68, 73, 74, 80, 81, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 121, 122, 136, 137, 144, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 196, 197, 202

Offset: 1

Reinhard Zumkeller, Aug 24 2008

nonn

A000161(a(n)) = 1.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A000548 is a subsequence.

A216283 Number of nonnegative solutions to the equation x^2+5*y^2 = n.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

Records occur at 1, 9, 81, 189, 441, 1449, 3969, 12789, 13041, 30429, ... - Antti Karttunen, Aug 23 2017

			For n = 9, there are two solutions: 9 = 3^2 + 5*(0^2) = 2^2 + 5*(1^2), thus a(9) = 2.
For n = 81, there are three solutions: 81  = 9^2 + 5*(0^2) = 6^2 + 5*(3^2) = 1^2 + 5*(4^2), thus a(81) = 3.

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

Cf. A033718 (all solutions x^2+5*y^2 = n).
Cf. A092573, A119395, A000161, A025426, A216282.
Cf. A020669 (positions of nonzeros).

N=666;  x='x+O('x^N);
T(x)=sum(n=0,ceil(sqrt(N)),x^(n*n));
Vec(T(x)*T(x^5))
/* Joerg Arndt, Sep 21 2012 */

(define (A216283 n) (cond ((< n 2) 1) (else (let loop ((k (A000196 n)) (s 0)) (if (< k 0) s (let ((x (- n (* k k)))) (loop (- k 1) (+ s (if (zero? (modulo x 5)) (A010052 (/ x 5)) 0))))))))) ;; Antti Karttunen, Aug 23 2017

G.f. T(x) * T(x^5) where T(x) = sum(n>=0, x^(n^2) ). - Joerg Arndt, Sep 21 2012

Examples from Antti Karttunen, Aug 23 2017

A247367 Number of ways to write n as a sum of a square and a nonsquare.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Sep 14 2014

nonn

			a(10) = #{0+10, 4+6} = 2;
a(11) = #{0+11, 1+10, 4+7, 9+2} = 4;
a(12) = #{0+12, 1+11, 4+8, 9+3} = 4;
a(13) = #{0+13, 1+12} = 2;
a(14) = #{0+14, 1+13, 4+10, 9+5} = 4;
a(15) = #{0+15, 1+14, 4+11, 9+6} = 4;
a(16) = #{1+15, 4+12, 9+7} = 3;
a(17) = #{0+17, 4+13, 9+8} = 3;
a(18) = #{0+18, 1+17, 4+14, 16+2} = 4;
a(19) = #{0+19, 1+18, 4+15, 9+10, 16+3} = 5;
a(20) = #{0+20, 1+19, 9+11} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000037, A000290, A010052, A000925, A000161.

a247367 n = sum $ map ((1 -) . a010052 . (n -)) $
                  takeWhile (<= n) a000290_list

sQ[n_] := sQ[n] = IntegerQ[Sqrt[n]];
a[n_] := Sum[Boole[sQ[k] && !sQ[n-k] || !sQ[k] && sQ[n-k]], {k, 0, Quotient[n, 2]}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 10 2022 *)

A321423 Expansion of 1/2 * Product_{0 <= i <= j} (1 + x^(i^2 + j^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 10, 11, 12, 12, 14, 16, 17, 20, 22, 25, 27, 29, 33, 36, 39, 41, 46, 52, 56, 60, 65, 72, 79, 84, 91, 100, 109, 118, 125, 136, 147, 158, 172, 185, 201, 215, 232, 251, 269, 287, 306, 331, 357, 381, 406, 436, 469, 503

Offset: 0

Seiichi Manyama, Nov 09 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000161, A321380, A321424.

G.f.: Product_{k>0} (1 + x^k)^A000161(k).

A125018 Numbers == 1 (mod 4) with a unique partition as a sum of 2 squares x^2 + y^2.

Original entry on oeis.org

1, 5, 9, 13, 17, 29, 37, 41, 45, 49, 53, 61, 73, 81, 89, 97, 101, 109, 113, 117, 121, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 333, 337, 349, 353, 361, 369, 373, 389, 397, 401, 405, 409, 421, 433

Offset: 1

Artur Jasinski, Nov 16 2006

nonn

			5 = 1^2 + 2^2, 9 = 0^2 + 3^2, 13 = 2^2 + 3^2, 17 = 1^2 + 4^2, 29 = 2^2 + 5^2, ... - _Michael Somos_, Jul 25 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002145, A119345, A124982, A124983.

Select[4 * Range[0, 100] + 1, Length @ PowersRepresentations[#, 2, 2] == 1 &] (* Amiram Eldar, Mar 12 2020 *)

isok(n)= {if (n % 4 != 1, return(0)); A000161(n) == 1;} \\ Michel Marcus, Nov 02 2013

More terms from Michel Marcus, Nov 02 2013

A133104 Number of partitions of n^4 into n nonzero squares.

Original entry on oeis.org

1, 0, 3, 1, 49, 732, 9659, 190169, 3225654, 61896383, 1360483727, 30969769918, 778612992660, 20749789703573, 579672756740101, 17115189938667708, 525530773660159970, 16825686497823918869, 561044904645283065043, 19368002907483932784642

Offset: 1

Hugo Pfoertner, Sep 11 2007

nonn

			a(3)=3 because there are 3 ways to express 3^4 = 81 as a sum of 3 nonzero squares: 81 = 1^2 + 4^2 + 8^2 = 3^2 + 6^2 + 6^2 = 4^2 + 4^2 + 7^2.
a(4)=1 because the only way to express 4^4 = 256 as a sum of 4 nonzero squares is 256 = 8^2 + 8^2 + 8^2 + 8^2.

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.

Cf. A133105 (number of ways to express n^4 as a sum of n distinct nonzero squares), A133103 (number of ways to express n^3 as a sum of n nonzero squares).

a(i, n, k)=local(s, j); if(k==1, if(issquare(n), return(1), return(0)), s=0; for(j=ceil(sqrt(n/k)), min(i, floor(sqrt(n-k+1))), s+=a(j, n-j^2, k-1)); return(s)) for(n=1,50, m=n^4; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

a(9) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

a(10) onwards from Robert Gerbicz, May 09 2008

A216278 Number of solutions to the equation x^2+2y^2 = n with x and y > 0.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

Cf. A092573, A119395, A000161, A025426.

r[n_] := Reduce[x > 0 && y > 0 && x^2 + 2 y^2 == n, Integers];
a[n_] := Which[rn = r[n]; rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1];
Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 24 2017 *)

A216279 Number of solutions to the equation x^2+5y^2 = n with x and y > 0.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A092573, A119395, A000161, A025426.

a(n)=sum(k=1,sqrtint((n-1)\5), issquare(n-5*k^2)) \\ Charles R Greathouse IV, Jun 06 2016

list(lim)=my(v=vector(lim\1),t); for(y=1,sqrtint((#v-1)\5), t=5*y^2; for(x=1,sqrtint(#v-t), v[x^2+t]++)); v \\ Charles R Greathouse IV, Jun 06 2016

A216280 Number of nonnegative solutions to the equation x^4 + y^4 = n.

Original entry on oeis.org

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

The first n with a(n) > 1 is 635318657 = 41 * 113 * 241 * 569, with a(635318657) = 2. Izadi, Khoshnam, & Nabardi show that for any n with a(n) > 1, the elliptic curve y^2 = x^3 - nx has rank at least 3. According to gp, y^2 = x^3 - 635318657x has analytic rank 4 (and first nonzero derivative around 35741.7839). - Charles R Greathouse IV, Jan 12 2017

Antti Karttunen, Table of n, a(n) for n = 1..16561
F. A. Izadi, F. Khoshnam, K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥4 (2012). arXiv:1202.5676 [math.NT], 2012.
F. A. Izadi, F. Khoshnam, K. Nabardi, Sums of two biquadrates and elliptic curves of rank ≥ 4, Math. J. Okayama Univ. 56 (2014), 51-63.

Cf. A025455, A025446, A000161, A025426, A216284.

Cf. A004831 (positions of nonzero terms).

Reap[For[n = 1, n <= 1000, n++, r = Reduce[0 <= x <= y && x^4 + y^4 == n, {x, y}, Integers]; sols = Which[r === False, 0, r[[0]] == And, 1, r[[0]] == Or, Length[r], True, Print[n, " ", r]]; If[sols != 0, Print[n, " ", sols, " ", r]]; Sow[sols]]][[2, 1]] (* Jean-François Alcover, Feb 22 2019 *)

a(n)=my(t=thue(thueinit('x^4+1,1),n)); sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1]) \\ Charles R Greathouse IV, Jan 12 2017

first(n)=my(T=thueinit('x^4+1,1),v=vector(n),t); for(k=1,n, t=thue(T,k); v[k]=sum(i=1,#t, t[i][1]>=0 && t[i][2]>=t[i][1])); v \\ Charles R Greathouse IV, Jan 12 2017

Offset added by Charles R Greathouse IV, Jan 12 2017

cp's OEIS Frontend

A133105 Number of partitions of n^4 into n distinct nonzero squares.

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A143575 Numbers m such that A143574(m) = m.

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A216283 Number of nonnegative solutions to the equation x^2+5*y^2 = n.

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A247367 Number of ways to write n as a sum of a square and a nonsquare.

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Haskell

Mathematica

A321423 Expansion of 1/2 * Product_{0 <= i <= j} (1 + x^(i^2 + j^2)).

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Formula

A125018 Numbers == 1 (mod 4) with a unique partition as a sum of 2 squares x^2 + y^2.

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Mathematica

PARI

Extensions

A133104 Number of partitions of n^4 into n nonzero squares.

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Examples

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Programs

PARI

Extensions

A216278 Number of solutions to the equation x^2+2y^2 = n with x and y > 0.

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Programs

Mathematica

A216279 Number of solutions to the equation x^2+5y^2 = n with x and y > 0.

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