A196224 Numbers n such that n^2 + n is not of the form x^2 + y^2 + z^2.

12, 15, 19, 44, 51, 63, 76, 83, 108, 112, 115, 140, 143, 147, 172, 179, 204, 211, 236, 240, 243, 255, 268, 271, 275, 300, 307, 332, 339, 364, 368, 371, 396, 399, 403, 428, 435, 448, 460, 467, 492, 496, 499, 524, 527, 531, 556, 563, 575, 588, 595, 620, 624

Offset

1,1

Comments

Nick Herbert calls these "Sirag Numbers" after Saul-Paul Sirag. Initially the idea arose by considering the quantum operators for spin or angular momentum, where J^2 = J[x]^2 + J[y]^2 + J[z]^2 = ħ^2 j(j+1), see link.

32n + 12 and 32n + 19 are members for all nonnegative n. All members are in {0, 12, 15, 16, 19, 31} mod 32. - Charles R Greathouse IV, Sep 29 2011

As noted in A004215, n is in the sequence iff n^2+n is of the form 4^i * (8*j+7).

Express J*(J+1) in base 4. If the last two nonzero digits are either 13 or 33, J is a Sirag number. - Jack Brennen, Sep 30 2011

n is in this sequence iff n == 12 or 19 (mod 32), n == 4^j*(8k+7), where j >= 2, or n == 4^j*(8k+1)-1, where j >= 2, k >= 0. - David W. Wilson, Oct 21 2011, (clarified by Mauro Fiorentini, May 11 2017)

Also n == 4^j - 1, where j >= 2 - Mauro Fiorentini, May 11 2017

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Nick Herbert, The Sirag Numbers, Aug 27 2011.

Wikipedia, Quantum numbers with spin-orbit interaction.

Formula

16/3 * n < a(n) < 16n. - Charles R Greathouse IV, Sep 29 2011

a(n) = 12n + O(log(n)). - David W. Wilson, Oct 21 2011

Mathematica

siragQ[n_]:=Module[{b4=IntegerDigits[n(n+1), 4]}, While[Last[b4]==0, b4= Drop[b4, -1]]; MemberQ[{{1, 3}, {3, 3}}, Take[b4, -2]]]; Select[Range[650], siragQ] (* Harvey P. Dale, relying on Jack Brennen's comment, Oct 01 2011 *)

Prog

(PARI) is_A196224(n)={ (n*=n+1)\4^valuation(n, 4)%8==7 } \\ M. F. Hasler, Sep 29 2011

Crossrefs

Cf. A004215, A002378.

Sequence in context: A350563 A120169 A115349 * A163657 A117815 A154390

Adjacent sequences: A196221 A196222 A196223 * A196225 A196226 A196227

Keyword

nonn

Author

M. F. Hasler, Sep 29 2011

Status

approved