A078134 -id:A078134 - cp's OEIS frontend

A092362 Number of partitions of n^2 into squares greater than 1.

1, 0, 1, 1, 2, 3, 5, 8, 11, 28, 44, 94, 167, 354, 643, 1314, 2412, 4792, 8981, 17374, 32566, 62008, 115702, 217040, 402396, 745795, 1372266, 2517983, 4595652, 8354350, 15125316, 27265107, 48972467, 87584837, 156119631, 277152178, 490437445, 864534950

Offset: 0

Reinhard Zumkeller, Mar 19 2004

nonn

a(n) = A078134(A000290(n)).

			a(6) = 5: 6^2 = 36 = 16+16+4 = 16+4+4+4+4+4 = 9+9+9+9 = 4+4+4+4+4+4+4+4+4.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A001156, A037444.

Cf. A093115, A093116.

b:=proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<2, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
   end:
a:= n-> b(n^2, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n^2, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3) / 2^(4/3)) * Zeta(3/2)^(4/3) / (2^(11/3) * sqrt(3) * Pi^(5/6) * n^(11/3)). - Vaclav Kotesovec, Apr 10 2017

Corrected a(0) and more terms from Alois P. Heinz, Apr 15 2013

A111178 Number of partitions of n into positive numbers one less than a square.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 2, 5, 2, 4, 5, 2, 5, 5, 2, 6, 7, 4, 6, 7, 5, 6, 8, 6, 8, 12, 6, 9, 13, 6, 10, 15, 8, 14, 15, 9, 16, 16, 10, 18, 21, 14, 19, 22, 16, 20, 24, 19, 25, 30, 20, 27, 33, 21, 29, 39, 26, 37, 40, 28, 42, 42, 31, 48

Offset: 0

Wouter Meeussen, Oct 22 2005

Also limiting form of the number of representations of n into k positive squares for k decreasing from n to 1, or Table[Count[SumOfSquaresRepresentations[k,n], {a_,}/;a>0], {n,100,100}, {k,100,40,-1}]. (Franklin T. Adams-Watters: replacing k^2 ones by the value k^2 changes the count by k^2-1).

a(n) = A243148(2n,n). - Alois P. Heinz, May 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

Cf. A001156, A078134.

Cf. A005563, A319799.

a111178 = p $ tail a005563_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 02 2014

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)+
      `if`(i^2>n+1, 0, b(n+1-i^2, i))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 30 2014

nn = 100; CoefficientList[Series[Product[1/(1 - x^(k^2 - 1)), {k, 2, nn}], {x, 0, nn}], x] (* corrected by T. D. Noe, Feb 22 2012 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n+1, 0, b[n+1-i^2, i]]]]; a[n_] := b[n, Round[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

G.f.: Product_{k>=2} 1/(1-x^(k^2-1)).

A134755 Minimal number such that all greater numbers can be written as sums of squares of primes in more than n ways.

Original entry on oeis.org

23, 39, 55, 64, 68, 73, 80, 84, 91, 96, 100, 105, 109, 113, 114, 118, 122, 123, 127, 131, 132, 136, 140, 140, 144, 145, 145, 149, 149, 153, 154, 156, 158, 160, 163, 164, 167, 168, 168, 172, 172, 176, 176, 176, 180, 180, 181, 181, 185, 185, 185, 189, 189, 190

Offset: 0

Hieronymus Fischer, Nov 11 2007

nonn

The sequence is well-defined, in that a(n) exists for all n>=0. Proof by induction: a(0) exists. We set b(j):=number of ways to write j as sum of squares of primes (=A090677). If a(n) exists, then b(j)>n for all j>a(n). Setting m:=a(n)+1, we find that there are n+1 sum of squares of primes B(0,i), 1<=i<=n+1, with m=B(0,i).
Further there are n+1 such sum expressions B(1,i), B(2,i) and B(3,i), 1<=i<=n+1, representing m+1, m+2 and m+3, respectively. For all j>a(n) we have j=m+4*floor((j-m)/4)+(j-m) mod 4. Thus j=m+r+s*2^2, where r=0,1,2 or 3. Hence n can be written B(r,i)+s*2^2 and there are n+1 such representations.
Let q be the maximal prime number (to be squared) occurring as a term within those sum expressions B(r,i), 0<=r<=3,1<=i<=n+1. We select a prime number p>q and we set c:=a(n)+p^2. For j>c, we have the n+1 representations B(r(j),i)+s(j)*2^2. Additionally, for j-p^2 (which is >a(n)) there are also n+1 representations B(r_p,i)+s_p*2^2, where r_p:=r(j-p^2), s_p:=s(j-p^2).
Thus j can be written B(r(j),i)+s(j)*2^2, 1<=i<=n+1 and B(r_p,i)+s_p*2^2+p^2, 1<=i<=n+1. By choice of p all these sum representations of j are different, which implies, that there are 2n+2 such representations. It follows b(j)>2n+2>n+1 for all j>c, which implies, that a(n+1) exists.

			a(0)=23, since numbers >23 can be written as sum of squares of primes.
a(1)=39, since there are at least two ways, to write a number >39 as a sum of squares of primes.

Cf. A078134, A078135, A078136, A078139, A090677, A078137, A134754.

a(n)=min( m | A090677(j)>n for all j>m).

A033183 a(n) = number of pairs (p,q) such that 4p + 9q = n.

Original entry on oeis.org

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

Offset: 0

Michel Tixier (tixier(AT)dyadel.net)

nonn

From Reinhard Zumkeller, Nov 07 2009: (Start)
In other words: number of partitions into 4 or 9;
a(n) <= A078134(n); a(A078135(n)) = 0;
a(A167632(n)) = n and a(m) < n for m < A167632(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1).

Cf. A033182.

CoefficientList[Series[1/((1-x^4)(1-x^9)),{x,0,80}],x] (* or  *) LinearRecurrence[{0,0,0,1,0,0,0,0,1,0,0,0,-1}, {1,0,0,0,1,0,0,0,1,1,0,0,1}, 80] (* Harvey P. Dale, Oct 13 2012 *)

a(n) = [ 7 n/9 ]+1+[ -3 n/4 ].
G.f.: 1/((1-x^4)*(1-x^9)). - Vladeta Jovovic, Nov 12 2004
a(n) = a(n-4) + a(n-9) - a(n-13). - R. J. Mathar, Dec 04 2011

A134754 Minimal number such that all greater numbers can be written as sums of squares >1 in more than n ways.

Original entry on oeis.org

23, 39, 39, 55, 55, 55, 59, 59, 63, 71, 71, 71, 71, 75, 75, 75, 75, 79, 79, 87, 87, 87, 87, 87, 91, 91, 91, 91, 91, 91, 95, 95, 95, 95, 95, 96, 96, 99, 99, 103, 103, 103, 103, 103, 103, 103, 107, 107, 107, 107, 107, 107, 107, 107, 111, 111, 111, 111, 111, 111, 111

Offset: 0

Hieronymus Fischer, Nov 11 2007

nonn

The sequence is well-defined, in that a(n) exists for all n>=0. For the reasoning see A078134.

			a(0)=23, since numbers >23 can be written as sum of squares >1.
a(2)=39, since there are at least three ways, to write a number >39 as a sum of squares >1.

Cf. A078134, A078135, A078136, A090677, A078137, A134755.

a(n)=min( m | A078134(j)>n for all j>m).

A093115 Number of partitions of n^2 into squares not greater than n.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 10, 13, 17, 108, 159, 228, 317, 430, 572, 748, 5753, 8125, 11266, 15376, 20672, 27430, 35942, 46575, 59717, 523905, 708028, 946875, 1253880, 1645224, 2140099, 2761318, 3535658, 4494602, 5674753, 7118724, 69766770, 90940578, 117756370

Offset: 0

Reinhard Zumkeller, Mar 21 2004

nonn

			n=6: 6^2 = 9*2^2 = 8*2^2+4*1^2 = 7*2^2+8*1^2 = 6*2^2+12*1^2 = 5*2^2+16*1^2 = 4*2^2+20*1^2 = 3*2^2+24*1^2 = 2*2^2+28*1^2 = 1*2^2+32*1^2 = 36*1^2, therefore a(6)=10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A093116, A092362, A001156, A037444, A078134.
Cf. A072925.
Cf. A072213, A161407. [Reinhard Zumkeller, Jun 10 2009]

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r-`if`(r^2>n, 1, 0))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n-i^2, i]]]]; a[n_] := (r = Sqrt[n] // Floor; b[n^2, r - If[r^2 > n, 1, 0]]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Coefficient of x^(n^2) in the series expansion of Product_{k=1..floor(sqrt(n))} 1/(1 - x^(k^2)). - Vladeta Jovovic, Mar 24 2004

More terms from Vladeta Jovovic, Mar 24 2004

Corrected a(0) by Alois P. Heinz, Apr 15 2013

A093116 Number of partitions of n^2 into squares not less than n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 2, 5, 4, 4, 5, 9, 15, 23, 24, 13, 20, 32, 55, 84, 113, 185, 303, 545, 167, 298, 435, 716, 1055, 1701, 2584, 4213, 6471, 10218, 15884, 4856, 7376, 11231, 17221, 26054, 39583, 60109, 91622, 138569, 209951, 318368, 483098, 730183

Offset: 0

Reinhard Zumkeller, Mar 21 2004

nonn

			n=10: 10^2 = 100 = 64+36 = 36+16+16+16+16 = 25+25+25+25, all other partitions of 100 into squares contain parts < 10, therefore a(10) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A093115, A092362, A001156, A037444, A078134.

Cf. A072213, A161408. [Reinhard Zumkeller, Jun 10 2009]

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i^2>n, 0, b(n, i+1) +b(n-i^2, i)))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r+`if`(r^2Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i^2>n, 0, b[n, i+1] + b[n-i^2, i]]]; a[n_] := With[{r = Sqrt[n]//Floor}, b[n^2, r + If[r^2Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A302833 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 19, 23, 27, 32, 38, 44, 50, 58, 67, 77, 87, 99, 112, 126, 140, 156, 175, 195, 216, 239, 265, 292, 320, 351, 385, 422, 460, 503, 549, 598, 648, 703, 763, 826, 892, 963, 1041, 1122, 1206, 1296, 1394, 1498, 1605, 1721, 1845, 1977, 2112, 2256, 2410, 2573

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Partial sums of A001156.

Number of partitions of n into squares if there are two kinds of 1's.

Cf. A000070, A000290, A001156, A078134, A279225, A302835.

b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 13 2018

nmax = 58; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 58; CoefficientList[Series[1/(1 - x) Sum[x^j^2/Product[(1 - x^k^2), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Sum_{j>=0} x^(j^2)/Product_{k=1..j} (1 - x^(k^2)).
From Vaclav Kotesovec, Apr 13 2018: (Start)
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2*Pi^(3/2) * sqrt(3*n)).
a(n) ~ 2^(4/3) * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A001156(n). (End)

A078138 Primes which can be written as sum of squares > 1.

Original entry on oeis.org

13, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311

Offset: 1

Reinhard Zumkeller, Nov 19 2002

By Sylvester's solution to the Frobenius problem, all integers greater than 4*9 - 4 - 9 = 23 can be represented as a sum of multiples of 4 and 9. Hence all primes except 2,3,5,7,11,19,23 are in this sequence. [Charles R Greathouse IV, Apr 19 2010]

			A000040(11) = 31 = 3^2 + 3^2 + 3^2 + 2^2, therefore 31 is a term.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. J. Sylvester, "Question 7382" in Mathematical Questions from the Educational Times, 37 (1884), p. 26 (search for 7382).
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Coin Problem
Index entries for sequences related to sums of squares

Cf. A078134, A078139, A078132.

Join[{13,17},Prime[Range[10,100]]] (* Harvey P. Dale, May 12 2014 *)

PARI
```
a(n)=if(n<3,[13,17][n],prime(n+7))
```

Comments, reference, and links by Charles R Greathouse IV, Apr 19 2010

A298640 Number of compositions (ordered partitions) of n^2 into squares > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 8, 12, 129, 874, 9630, 167001, 3043147, 72844510, 2423789655, 106665874384, 6156805673648, 470151743582651, 47558937432498729, 6363358599941131580, 1126147544855148769425, 263646401550138303553708, 81649922556593759124887197

Offset: 0

Ilya Gutkovskiy, Jan 24 2018

nonn

			a(5) = 8 because we have [25], [16, 9], [9, 16], [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].

Cf. A000290, A006456, A078134, A224366, A280542, A298642.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j^2), j=2..isqrt(n)))
    end:
a:= n-> b(n^2):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 05 2018

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j^2], {j, 2, Floor @ Sqrt[n]}]];
a[n_] := b[n^2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

a(n) = [x^(n^2)] 1/(1 - Sum_{k>=2} x^(k^2)).

a(n) = A280542(A000290(n)).

cp's OEIS Frontend

A092362 Number of partitions of n^2 into squares greater than 1.

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A111178 Number of partitions of n into positive numbers one less than a square.

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A134755 Minimal number such that all greater numbers can be written as sums of squares of primes in more than n ways.

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A033183 a(n) = number of pairs (p,q) such that 4*p + 9*q = n.

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A134754 Minimal number such that all greater numbers can be written as sums of squares >1 in more than n ways.

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A093115 Number of partitions of n^2 into squares not greater than n.

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Maple

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A093116 Number of partitions of n^2 into squares not less than n.

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Maple

Mathematica

A302833 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)).

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A033183 a(n) = number of pairs (p,q) such that 4p + 9q = n.