A002481 Numbers of form x^2 + 6y^2.

0, 1, 4, 6, 7, 9, 10, 15, 16, 22, 24, 25, 28, 31, 33, 36, 40, 42, 49, 54, 55, 58, 60, 63, 64, 70, 73, 79, 81, 87, 88, 90, 96, 97, 100, 103, 105, 106, 112, 118, 121, 124, 127, 132, 135, 144, 145, 150, 151, 154, 159, 160, 166, 168, 169, 175, 177, 186, 193, 196, 198, 199, 202, 214

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Norms of numbers in Z[sqrt(-6)]. - Alonso del Arte, Sep 23 2014

It seems that a positive integer n is in this sequence if and only if the p-adic order ord_p(n) of n is even for any prime p with floor(p/12) odd, and the number of prime divisors p == 5 or 11 (mod 24) with ord_p(n) odd has the same parity with ord_2(n) + ord_3(n). - Zhi-Wei Sun, Mar 24 2018

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Zak Seidov, Table of n, a(n) for n = 1..2064 (terms <= 10000).
• Leonhard Euler, E388 Vollständige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 425.

Crossrefs

Cf. A020669, A033199.

Programs

Maple

N:= 10^4: # to get all terms <= N
{seq(seq(a^2 + 6*b^2, a = 0 .. floor(sqrt(N-6*b^2))), b = 0 .. floor(sqrt(N/6)))};
# for Maple 11, or earlier, uncomment the next line
# sort(convert(%,list));  # Robert Israel, Sep 24 2014

Mathematica

lim = 10^4; k = 6; Union@Flatten@Table[x^2 + k * y^2, {y, 0, Sqrt[lim/k]}, {x, 0, Sqrt[lim - k * y^2]}] (* Zak Seidov, Mar 30 2011 *)