A316489 -id:A316489 - cp's OEIS frontend

A316490 Smallest positive number of the form 8*k + 4 that can be expressed as the sum of four distinct odd squares in exactly n ways, or 0 if no such number exists.

4, 84, 156, 260, 380, 596, 420, 588, 732, 884, 660, 876, 900, 1164, 924, 1236, 1140, 1452, 1428, 1524, 1380, 1260, 1620, 2060, 1596, 1764, 1740, 2196, 2364, 2628, 1980, 3236, 2244, 2676, 2220, 2100, 2460, 3916, 2844, 2916, 2580, 2340, 2700, 4532, 3396, 4300

Offset: 0

Jon E. Schoenfield, Jul 28 2018

nonn

Every odd square is a number of the form 8*k + 1, so every sum of four odd squares is a number of the form 8*k + 4.
A316489 lists all positive numbers of the form 8*k + 4 that cannot be expressed as the sum of four distinct odd squares.
A316834 lists all numbers that can be expressed in only one way as the sum of four distinct odd squares.
If a(571) > 0 then a(571) > 4*10^6. For 48 values of 0 <= n <= 9999 a(n) = 0 or at least 4*10^6. - David A. Corneth, Nov 28 2020

			The smallest positive number of the form 8*k + 4 is 4, which cannot be expressed as the sum of four distinct odd squares, so a(0) = 4.
The smallest number that can be expressed as the sum of four distinct odd squares is the sum of the first four odd squares, i.e., 1^2 + 3^2 + 5^2 + 7^2 = 84, which cannot be so expressed in any other way, so a(1) = 84.
a(6) = 420 because 420 is the smallest number that can be expressed as the sum of four distinct odd squares in exactly 6 ways:
  420 =  1^2 +  3^2 +  7^2 + 19^2
      =  1^2 +  3^2 + 11^2 + 17^2
      =  1^2 +  5^2 + 13^2 + 15^2
      =  1^2 +  7^2 +  9^2 + 17^2
      =  5^2 +  7^2 + 11^2 + 15^2
      =  7^2 +  9^2 + 11^2 + 13^2.
(Sums such as 3^2 + 5^2 + 5^2 + 19^2 are not counted because the four odd squares are not all distinct.)

David A. Corneth, Table of n, a(n) for n = 0..570 (first 501 terms from Alois P. Heinz)
David A. Corneth, a(0)..a(9999) for values <= 4*10^6

Cf. A316489 (0 ways), A316834 (1 way).

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(min(i, t)<1, 0, b(n, i-2, t)+
      `if`(i^2>n, 0, b(n-i^2, i-2, t-1))))
    end:
a:= proc(n) option remember; local k;
      for k from 4 by 8 while n <> b(k, (r->
      r+1-irem(r, 2))(isqrt(k)), 4) do od; k
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 07 2018

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[Min[i, t] < 1, 0, b[n, i - 2, t] + If[i^2 > n, 0, b[n - i^2, i - 2, t - 1]]]];
a[n_] := a[n] = Module[{k}, For[k = 4, n != b[k, # + 1 - Mod[#, 2]& @ Floor @ Sqrt[k], 4], k += 8]; k];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

A316491 Number of ways to represent 8*n + 4 as the sum of four distinct odd squares.

Original entry on oeis.org

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

Offset: 0

Jon E. Schoenfield, Jul 29 2018

nonn

Every odd square is a number of the form 8*k + 1, so every sum of four odd squares is a number of the form 8*k + 4.
A316489 lists all positive numbers of the form 8*k + 4 that cannot be expressed as the sum of four distinct odd squares; for each such number, a(k)=0.
A316834 lists all numbers that can be expressed in only one way as the sum of four distinct odd squares; each such number is of the form 8*k + 4, and for each such number, a(k)=1.

			n=1: 8*1 + 4 = 12 cannot be expressed as the sum of four distinct odd squares, so a(1)=0.
n=10: 8*10 + 4 = 84 can be expressed as the sum of four distinct odd squares in only 1 way (84 = 1^2 + 3^2 + 5^2 + 7^2), so a(10)=1.
n=19: 8*19 + 4 = 156 can be expressed as the sum of four distinct odd squares in exactly 2 ways (156 = 1^2 + 3^2 + 5^2 + 11^2 = 1^2 + 5^2 + 7^2 + 9^2), so a(19)=2.

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

Cf. A316834, A316489, A316490.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(min(i, t)<1, 0, b(n, i-2, t)+
      `if`(i^2>n, 0, b(n-i^2, i-2, t-1))))
    end:
a:= n-> (m-> b(m, (r-> r+1-irem(r, 2))(isqrt(m)), 4))(8*n+4):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 05 2018

a[n_] := Count[ IntegerPartitions[8 n + 4, {4}, Range[1, Sqrt[8 n + 4], 2]^2], w_ /; Max@Differences@w < 0]; Array[a, 87, 0] (* Giovanni Resta, Aug 12 2018 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0],
     If[Min[i, t] < 1, 0, b[n, i-2, t] +
     If[i^2 > n, 0, b[n-i^2, i-2, t-1]]]];
a[n_] := Function[m, b[m, Function[r, r+1-Mod[r, 2]][Floor@Sqrt[m]], 4]][8n+4];
a /@ Range[0, 100] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A316490 Smallest positive number of the form 8*k + 4 that can be expressed as the sum of four distinct odd squares in exactly n ways, or 0 if no such number exists.

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