A025428 -id:A025428 - cp's OEIS frontend

A375731 a(n) is the number of partitions of n having a square number of parts whose sum of squares is a square.

1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 4, 2, 2, 7, 4, 4, 7, 7, 6, 9, 12, 9, 21, 21, 19, 26, 30, 32, 43, 54, 54, 64, 87, 85, 119, 128, 146, 174, 205, 213, 281, 324, 368, 420, 503, 531, 688, 760, 837, 992, 1174, 1252, 1535, 1705, 1931, 2236, 2619, 2821, 3402, 3769, 4272

Offset: 0

Felix Huber, Aug 28 2024

nonn

			a(13) counts the 4 partitions [1, 1, 1, 1, 1, 1, 1, 3, 3] with 9 = 3^2 parts and 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 3^2 = 5^2, [1, 4, 4, 4] with 2^2 parts and 1^2 + 4^2 + 4^2 + 4^2 = 7^2, [2, 2, 4, 5] with 4 = 2^2 parts and 2^2 + 2^2 + 4^2 + 5^2 = 7^2, [13] with 1 = 1^2 part and 13^2 = 13^2.

Cf. A001156, A025428, A006456, A037444, A240127, A240128, A375732.

# first Maple program to calculate the sequence:
A375731:=proc(n) local a,i,j; a:=0; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then a:=a+1 fi od; return a end proc; seq(A375731(n),n=0..63);
# second Maple program to calculate the partitions:
A375731part:=proc(n) local L,i,j;L:=[]; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then L:=[op(L),i] fi od; return op(L) end proc; A375731part(13);

a(n) = my(nb=0); forpart(p=n, if (issquare(#p) && issquare(norml2(Vec(p))), nb++)); nb; \\ Michel Marcus, Aug 30 2024

1 <= a(n) <= A240127(n).

A293175 Integers with precisely six partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

66, 81, 97, 99, 105, 110, 115, 121, 123, 124, 137, 139, 141, 149, 155, 156, 158, 159, 164, 179, 188, 239, 264, 284, 440, 496, 624, 632, 656, 752, 1056, 1136, 1760, 1984, 2496, 2528, 2624, 3008, 4224, 4544, 7040, 7936, 9984, 10112, 10496, 12032, 16896, 18176

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 6.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297.

f[n_] := Length@ PowersRepresentations[n, 4, 2]; Select[ Range@ 19000, f@# == 6 &] (* Robert G. Wilson v, Oct 27 2017 *)

A294308 Integers with precisely seven partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

82, 98, 100, 102, 106, 108, 118, 125, 129, 132, 133, 134, 135, 161, 163, 173, 183, 197, 199, 204, 211, 212, 215, 236, 263, 328, 392, 400, 408, 424, 432, 472, 528, 536, 816, 848, 944, 1312, 1568, 1600, 1632, 1696, 1728, 1888, 2112, 2144, 3264, 3392, 3776

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 7.

Robert Price, Table of n, a(n) for n = 1..74
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297, A293175.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@650, f@#==7 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294310 Integers with precisely nine partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

90, 146, 166, 174, 185, 187, 205, 206, 207, 209, 219, 220, 221, 223, 231, 235, 251, 260, 271, 287, 316, 359, 360, 380, 584, 664, 696, 824, 880, 1040, 1264, 1440, 1520, 2336, 2656, 2784, 3296, 3520, 4160, 5056, 5760, 6080, 9344, 10624, 11136, 13184, 14080

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 9.

Robert Price, Table of n, a(n) for n = 1..60
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==9 &] (* Vincenzo Librandi, Oct 28 2017 *)

A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

Original entry on oeis.org

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

Offset: 0

Felix Huber, Jul 22 2025

nonn

a(n) is the number of partitions of n into 4 nonzero squares < n/2.

			The a(51) = 1 multiset is [1, 9, 16, 25].
The a(52) = 3 multisets are [1, 1, 25, 25], [4, 16, 16, 16] and [9, 9, 9, 25].

Felix Huber, Table of n, a(n) for n = 0..10000

Cf. A000290, A000414, A000534, A025428, A062890, A385736.

# After Alois P. Heinz (A025428)
b:=proc(n,i,t)
    option remember;
    `if`(n=0,`if`(t=0,1,0),`if`(i<1 or t<1, 0, b(n,i-1,t)+`if`(i^2>n,0,b(n-i^2,i,t-1))))
    end:
A385860:=n->b(n,floor(sqrt((n-1)/2)),4):
seq(A385860(n),n=0..87);

a(n) <= A025428(n).

A025769 Expansion of 1/((1-x)(1-x^3)(1-x^8)).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 30, 32, 33, 35, 37, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 61, 63, 65, 68, 70, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 99, 102, 105, 108, 111

Offset: 0

N. J. A. Sloane

Mark Underwood observed that the number of partitions into four nonzero squares of the squares of primes is given by A025428(A001248(n)) = a(prime(n)-4), cf. sequence A216374. - M. F. Hasler, Sep 16 2012
a(n) is the number of partitions of n into parts 1, 3, and 8. - Joerg Arndt, Apr 05 2024
a(n) = a(-12-n) for all n in Z using the floor definition. - Michael Somos, Apr 04 2024

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ... - _Michael Somos_, Apr 04 2024

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,0,0,1,-1,0,-1,1).

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^8)),{x,0,60}],x] (* or *) LinearRecurrence[{1,0,1,-1,0,0,0,1,-1,0,-1,1},{1,1,1,2,2,2,3,3,4,5,5,6},60] (* Harvey P. Dale, Nov 29 2012 *)
a[ n_] := Floor[((n+6)^2/16 + 1)/3]; (* Michael Somos, Apr 04 2024 *)

A025769(n)=((n+6)^2+16)\48  \\ M. F. Hasler, Sep 16 2012

a(n) = floor((x^2+12x+c)/48) with 51 <= c <= 58. - M. F. Hasler, Sep 16 2012

a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=2, a(5)=2, a(6)=3, a(7)=3, a(8)=4, a(9)=5, a(10)=5, a(11)=6, a(n)=a(n-1)+a(n-3)-a(n-4)+a(n-8)-a(n-9)- a(n-11)+ a(n-12). - Harvey P. Dale, Nov 29 2012

A218491 Number of ways that prime(n) can be represented as the sum of four nonzero squares.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 2, 2, 0, 2, 2, 1, 1, 2, 3, 2, 3, 3, 2, 1, 4, 1, 5, 2, 3, 2, 5, 2, 3, 4, 2, 5, 6, 6, 5, 3, 3, 5, 5, 6, 4, 7, 5, 9, 5, 7, 4, 6, 6, 5, 5, 7, 4, 9, 8, 4, 9, 6, 10, 8, 10, 7, 9, 9, 7, 9, 8, 9, 13, 10, 10, 11, 7, 13, 7, 10, 8, 11, 10, 13

Offset: 1

Arkadiusz Wesolowski, Nov 02 2012

nonn

a(pi(A213721(n))) = n, where pi(n) is the prime counting function.

			a(11) = 2 because prime(11) = 31 = 2*1 + 4 + 25 = 4 + 3*9.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares

Cf. A025428, A213721.

Table[Count[PowersRepresentations[Prime[n], 4, 2], _?(Min[#] > 0 &)], {n, 84}]

A223726 Multiplicities for A004433: sum of four distinct nonzero squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 3, 2, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 1, 2, 4, 2, 1, 2, 3, 1, 5, 2, 2, 2, 2, 3, 4, 3, 1, 4, 1, 1, 4, 2, 2, 2, 5, 3, 1, 6, 3, 3, 1, 2, 1, 1, 4, 4, 1, 2, 5, 1, 3, 7, 3, 2, 3, 4

Offset: 1

Wolfdieter Lang, Mar 26 2013

nonn

The number A004433(n) can be partitioned into four distinct parts, each of which is a nonzero square, and a(n) gives the multiplicity which is the number of different partitions of this type.

			a(1) = 1 because  A004433(1) = 30 has only one representation as sum of four distinct nonzero squares, given by the quadruple [1,2,3,4]: 1^2 + 2^2 + 3^2 + 4^2 = 30.
a(16) = 3 because for A004433(3) = 78 the three different quadruples are [1, 2, 3, 8], [1, 4, 5, 6] and [2, 3, 4, 7].
a(48) = 5 because A004433(48) = 126 has five different  representations given by the five quadruples [1, 3, 4, 10], [1, 5, 6, 8], [2, 3, 7, 8], [2, 4, 5, 9], [4, 5, 6, 7].

Cf. A004433, A025428, A000414, A097203, A222949.

a(n) = k if there are k different quadruples [s(1),s(2),2(3),s(4)] with increasing positive entries with sum(s(j)^2,j=1..4) = A004433(n), n >= 1.

A294309 Integers with precisely eight partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

114, 117, 122, 126, 145, 147, 148, 157, 165, 169, 172, 175, 177, 181, 190, 193, 203, 227, 233, 311, 456, 488, 504, 592, 688, 760, 1824, 1952, 2016, 2368, 2752, 3040, 7296, 7808, 8064, 9472, 11008, 12160, 29184, 31232, 32256, 37888, 44032, 48640

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 8.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==8 &] (* Vincenzo Librandi, Oct 28 2017 *)

A294311 Integers with precisely ten partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

130, 138, 153, 154, 171, 180, 182, 195, 196, 201, 213, 214, 217, 228, 229, 238, 241, 244, 247, 249, 253, 254, 257, 259, 269, 276, 277, 281, 295, 299, 303, 308, 317, 319, 332, 335, 347, 428, 431, 520, 552, 616, 720, 728, 784, 856, 912, 952, 976, 1016, 1104

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 10.

Robert Price, Table of n, a(n) for n = 1..98
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282, A294297, A293175, A294308, A294309, A294310.

f[n_]:=Length@PowersRepresentations[n, 4, 2]; Select[Range@850, f@#==10 &] (* Vincenzo Librandi, Oct 28 2017 *)

cp's OEIS Frontend

A375731 a(n) is the number of partitions of n having a square number of parts whose sum of squares is a square.

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A293175 Integers with precisely six partitions into sums of four squares of nonnegative numbers.

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A294308 Integers with precisely seven partitions into sums of four squares of nonnegative numbers.

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A294310 Integers with precisely nine partitions into sums of four squares of nonnegative numbers.

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A385860 a(n) is the number of distinct multisets of sides of quadrilaterals with perimeter n, where all four sides are squares.

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A025769 Expansion of 1/((1-x)*(1-x^3)*(1-x^8)).

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Mathematica

PARI

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A218491 Number of ways that prime(n) can be represented as the sum of four nonzero squares.

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Mathematica

A223726 Multiplicities for A004433: sum of four distinct nonzero squares.

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Examples

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Formula

A294309 Integers with precisely eight partitions into sums of four squares of nonnegative numbers.

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A025769 Expansion of 1/((1-x)(1-x^3)(1-x^8)).