cp's OEIS Frontend

A379009 a(n) = 2*n^2 XOR 1+sigma(n^2).

%I A379009 #17 Dec 19 2024 06:20:08
%S A379009 0,0,28,0,18,20,88,0,216,18,116,180,490,24,86,0,886,472,940,226,404,
%T A379009 108,1544,756,2028,74,500,200,1530,3086,1120,0,3648,366,3962,1160,
%U A379009 3890,292,686,994,2974,6540,2324,7996,378,8104,6544,3060,6192,1748,7114,778,7874,2860,1982,1224,2616,3482,5860,11502,5082
%N A379009 a(n) = 2*n^2 XOR 1+sigma(n^2).
%C A379009 For any hypothetical quasiperfect number q^2 (for which sigma(q^2) = 2*q^2 + 1, which are known to be odd squares if they exist at all, see references in A336701), a(q) would be equal to 2*q^2 XOR 2*(q^2)+2 = 2*(q^2 XOR q^2+1) = 2*A038712(1+q^2) = 2*3 = 6.
%C A379009 a(n) = 0 if n^2 is a square that is "almost perfect", also known as "least deficient". Only known examples are powers of 2. See A000079, A033879.
%H A379009 Antti Karttunen, <a href="/A379009/b379009.txt">Table of n, a(n) for n = 1..32768</a>
%H A379009 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A379009 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A379009 a(n) = A378988(A000290(n)).
%t A379009 Map[BitXor[2*#, DivisorSigma[1, #] + 1] &, Range[100]^2] (* _Paolo Xausa_, Dec 18 2024 *)
%o A379009 (PARI) A379009(n) = bitxor(2*(n^2),1+sigma(n^2));
%Y A379009 Cf. A000079 (conjectured to give positions of all 0's), A000290, A003987, A033879, A065764, A336701, A378988.
%Y A379009 Cf. also A378999, A379007.
%K A379009 nonn
%O A379009 1,3
%A A379009 _Antti Karttunen_, Dec 16 2024