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%I #55 May 11 2017 13:51:24
%S 12,15,19,44,51,63,76,83,108,112,115,140,143,147,172,179,204,211,236,
%T 240,243,255,268,271,275,300,307,332,339,364,368,371,396,399,403,428,
%U 435,448,460,467,492,496,499,524,527,531,556,563,575,588,595,620,624
%N Numbers n such that n^2 + n is not of the form x^2 + y^2 + z^2.
%C 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.
%C 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
%C As noted in A004215, n is in the sequence iff n^2+n is of the form 4^i * (8*j+7).
%C 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
%C 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)
%C Also n == 4^j - 1, where j >= 2 - _Mauro Fiorentini_, May 11 2017
%H Charles R Greathouse IV, <a href="/A196224/b196224.txt">Table of n, a(n) for n = 1..10000</a>
%H Nick Herbert, <a href="http://quantumtantra.blogspot.com/2011/08/sirag-numbers.html">The Sirag Numbers</a>, Aug 27 2011.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Quantum_number#Quantum_numbers_with_spin-orbit_interaction">Quantum numbers with spin-orbit interaction</a>.
%F 16/3 * n < a(n) < 16n. - _Charles R Greathouse IV_, Sep 29 2011
%F a(n) = 12n + O(log(n)). - _David W. Wilson_, Oct 21 2011
%t 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 *)
%o (PARI) is_A196224(n)={ (n*=n+1)\4^valuation(n,4)%8==7 } \\ _M. F. Hasler_, Sep 29 2011
%Y Cf. A004215, A002378.
%K nonn
%O 1,1
%A _M. F. Hasler_, Sep 29 2011
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