A025417 -id:A025417 - cp's OEIS frontend

A025443 Number of partitions of n into 4 distinct nonzero squares.

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

Offset: 0

David W. Wilson

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

Cf. A025428 (not necessarily distinct), A025376-A025394 (subsequences), A025417 (greedy inverse).

Column k=4 of A341040.

b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
      `if`(t*i^2n, 0, b(n-i^2,i-1,t-1))))
    end:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..150);  # Alois P. Heinz, Feb 07 2013

b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[t*i^2n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n, Sqrt[n] // Floor, 4]; Table[a[n], {n, 0, 150}] (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz*)
dnzs[n_]:=Length[Select[IntegerPartitions[n,{4}],Length[Union[#]]==4&&AllTrue[ Sqrt[ #], IntegerQ] && FreeQ[#,0]&]]; Array[dnzs,110,0] (* Harvey P. Dale, Jun 09 2024 *)

a(n) = [x^n y^4] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019

A350241 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

1, 65, 101, 142, 175, 255, 316, 380, 501, 625, 794, 995, 1155, 1456, 1696, 2012, 2373, 2709, 3118, 3566, 4158, 4608, 5211, 5852, 6500, 7221, 8065, 8906, 9766, 11089, 11855, 12868, 14020, 15337, 16601, 17854, 19255, 20840, 22364, 23964, 25813, 27665, 29650, 31635

Offset: 1

Ilya Gutkovskiy, Dec 21 2021

nonn

			For n = 2: 65 = 1^2 + 8^2 = 4^2 + 7^2.
For n = 3: 101 = 1^2 + 6^2 + 8^2 = 2^2 + 4^2 + 9^2 = 4^2 + 6^2 + 7^2.

Index entries for sequences related to sums of squares

Cf. A000290, A025341, A025379, A025415, A025417, A052261, A093195, A097563, A215908, A341040.

More terms from Jinyuan Wang, Dec 21 2021

A374228 a(n) is the smallest number which can be represented as the sum of four distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

10, 90, 1521, 300834

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 1521 = 1^3 + 2^3 + 8^3 + 10^3 = 1^3 + 4^3 + 5^3 + 11^3 = 4^3 + 6^3 + 8^3 + 9^3.

Cf. A025417, A025421, A146756, A374226, A374227.

A214513 Least number having n orderless representations as p^2 + q^2 + r^2 + s^2, where p, q, r, and s are primes.

Original entry on oeis.org

16, 148, 196, 436, 388, 628, 868, 988, 1228, 1468, 1708, 2212, 2068, 2860, 2620, 2380, 3220, 3388, 3700, 4108, 3940, 4180, 5260, 4228, 5068, 4900, 5500, 6220, 6340, 7780, 5908, 5740, 6580, 7540, 8260, 7420, 8860, 9340, 11260, 10708, 9940, 9100, 10180, 12820

Offset: 1

T. D. Noe, Jul 29 2012

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A025416, A025417, A214511, A214512.

nn = 10^5; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2 + ps[[k]]^2 + ps[[l]]^2, {i, Length[ps]}, {j, i, Length[ps]}, {k, j, Length[ps]}, {l, k, Length[ps]}]]; t = Select[t, # <= nn &]; t2 = Sort[Tally[t]]; u = Union[Transpose[t2][[2]]]; d = Complement[Range[u[[-1]]], u]; If[d == {}, nLim = u[[-1]], nLim = d[[1]]-1]; t3 = Table[Select[t2, #[[2]] == n &, 1][[1]], {n, nLim}]; Transpose[t3][[1]]

A374696 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

354, 6834, 16578, 300834, 2147874, 3847554, 16408434, 13155858, 489597858, 677125218, 780595299, 2374692243, 803898018, 5645172978

Offset: 1

Ilya Gutkovskiy, Jul 17 2024

			a(2) = 6834 = 1^4 + 2^4 + 4^4 + 9^4 = 3^4 + 4^4 + 7^4 + 8^4.
a(3) = 16578 = 1^4 + 2^4 + 9^4 + 10^4 = 2^4 + 5^4 + 6^4 + 11^4 = 3^4 + 7^4 + 8^4 + 10^4.

Cf. A025417, A025421, A230562, A374693.

a(9)-a(14) from Michael S. Branicky, Jul 21 2024

A374274 a(n) is the smallest number which can be represented as the sum of four distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

47, 142, 344, 652, 834, 1542, 2263, 3090, 4792, 4570, 5844, 8480, 9571, 10542, 15892, 18202, 19587, 23166, 26732, 32040, 36371, 39730, 44709, 52940, 55141, 60362, 67705, 79624, 86906, 89266, 103591, 116246, 126610, 131462, 135324, 148190, 158152, 162422, 186126, 200254

Offset: 3

Ilya Gutkovskiy, Jul 02 2024

nonn

			a(3) = 47 = 1 + 3 + 15 + 28 = 1 + 10 + 15 + 21 = 3 + 6 + 10 + 28.
a(4) = 142 = 1^2 + 2^2 + 4^2 + 11^2 = 1^2 + 4^2 + 5^2 + 10^2 = 2^2 + 5^2 + 7^2 + 8^2 = 3^2 + 4^2 + 6^2 + 9^2.

Michael S. Branicky, Table of n, a(n) for n = 3..204
Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A025417, A057145, A350405, A374144, A374273.

a(36) and beyond from Michael S. Branicky, Jul 08 2024

A374807 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero triangular numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

20, 38, 47, 64, 73, 92, 97, 110, 127, 115, 130, 164, 185, 172, 208, 157, 199, 235, 247, 232, 220, 272, 277, 304, 280, 361, 262, 307, 319, 391, 322, 292, 495, 415, 337, 367, 370, 382, 478, 482, 412, 409, 445, 430, 467, 500, 427, 532, 493

Offset: 1

Ilya Gutkovskiy, Jul 20 2024

nonn

			a(3) = 47 = 1 + 3 + 15 + 28 = 1 + 10 + 15 + 21 = 3 + 6 + 10 + 28.

Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A025417, A342326, A374806.

A374803 a(n) is the smallest number which can be represented as the sum of 4 distinct positive fifth powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

1300, 4062500, 1479604544

Offset: 1

Ilya Gutkovskiy, Jul 20 2024

			a(2) = 4062500 = 1^5 + 14^5 + 16^5 + 19^5 = 5^5 + 10^5 + 15^5 + 20^5.
a(3) = 1479604544 = 3^5 + 48^5 + 52^5 + 61^5 = 13^5 + 36^5 + 51^5 + 64^5 = 18^5 + 36^5 + 44^5 + 66^5.

Cf. A025417, A025421, A374696, A374804.

A374804 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero sixth powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

4890, 160426515, 1885800643779

Offset: 1

Ilya Gutkovskiy, Jul 20 2024

			a(2) = 160426515 = 1^6 + 3^6 + 19^6 + 22^6 = 1^6 + 10^6 + 15^6 + 23^6.
a(3) = 1885800643779 = 1^6 + 34^6 + 49^6 + 111^6 = 7^6 + 43^6 + 69^6 + 110^6 = 18^6 + 25^6 + 77^6 + 109^6.

Cf. A025417, A025421, A374696, A374803.

cp's OEIS Frontend

A025443 Number of partitions of n into 4 distinct nonzero squares.

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A350241 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly n ways, or 0 if no such number exists.

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A374228 a(n) is the smallest number which can be represented as the sum of four distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A214513 Least number having n orderless representations as p^2 + q^2 + r^2 + s^2, where p, q, r, and s are primes.

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A374696 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero fourth powers in exactly n ways, or -1 if no such number exists.

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A374274 a(n) is the smallest number which can be represented as the sum of four distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A374807 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero triangular numbers in exactly n ways, or -1 if no such number exists.

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A374803 a(n) is the smallest number which can be represented as the sum of 4 distinct positive fifth powers in exactly n ways, or -1 if no such number exists.

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A374804 a(n) is the smallest number which can be represented as the sum of 4 distinct nonzero sixth powers in exactly n ways, or -1 if no such number exists.

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