A200977 -id:A200977 - cp's OEIS frontend

A374285 a(n) is the smallest nonnegative integer k where there are exactly n nonnegative integer solutions to x^2 + 2*y^2 = k.

5, 0, 9, 81, 297, 891, 1683, 8019, 5049, 18513, 15147, 88209, 31977, 1185921, 136323, 166617, 95931, 7144929, 351747

Offset: 0

Seiichi Manyama, Jul 02 2024

For n > 1, a(n) is the smallest nonnegative k such that A216282(k) = n.
a(20) = 287793.
a(21) = 1499553.
a(22) = 11042163.
a(23) = 2832489.
a(24) = 863379.
a(25) = 20160657.
a(26) = 99379467.
a(27) = 5979699.
a(28) = 2590137.
a(30) = 3165723.
a(32) = 3933171.
a(33) = 121463793.
a(35) = 181445913.
a(36) = 9497169.

Cf. A374286, A374288.

Cf. A200977, A216282, A374294.

b(n, k) = sum(i=0, sqrtint(n), sum(j=0, sqrtint(n\k), i^2+k*j^2==n));
a(n, k=2) = my(cnt=0); while(b(cnt, k)!=n, cnt++); cnt;

A374294 a(n) is the smallest positive integer k such that A002325(k) = n.

Original entry on oeis.org

1, 3, 9, 27, 81, 99, 729, 297, 1089, 891, 59049, 1683, 531441, 8019, 9801, 5049, 43046721, 18513, 387420489, 15147, 88209, 649539, 31381059609, 31977, 1185921, 5845851, 314721, 136323, 22876792454961, 166617, 205891132094649, 95931, 7144929, 473513931, 10673289, 351747, 150094635296999121

Offset: 1

Seiichi Manyama, Jul 02 2024

nonn

			   n |  a(n)
-----+-----------------------
   2 |     3.
   3 |     9 = 3^2.
   4 |    27 = 3^3.
   5 |    81 = 3^4.
   6 |    99 = 3^2 * 11.
   7 |   729 = 3^6.
   8 |   297 = 3^3 * 11.
   9 |  1089 = 3^2 * 11^2.
  10 |   891 = 3^4 * 11.
  11 | 59049 = 3^10.
  12 |  1683 = 3^2 * 11 * 17.

Cf. A002325, A033715, A033200, A200977, A374295.

If p is prime, a(p) = 3^(p-1).

a(n) is divisible by 3 for n > 1.

cp's OEIS Frontend

A374285 a(n) is the smallest nonnegative integer k where there are exactly n nonnegative integer solutions to x^2 + 2*y^2 = k.

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A374294 a(n) is the smallest positive integer k such that A002325(k) = n.

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