A379007 -id:A379007 - cp's OEIS frontend

A283750 a(n) = n^2 XOR (n + 1)^2.

1, 5, 13, 25, 9, 61, 21, 113, 17, 53, 29, 233, 57, 109, 37, 481, 33, 101, 45, 249, 41, 93, 1013, 81, 49, 213, 125, 457, 89, 205, 69, 1985, 65, 197, 77, 473, 73, 253, 85, 945, 209, 117, 477, 169, 121, 4013, 229, 417, 97, 165, 1005, 185, 105, 413, 181, 1937, 241, 405, 189, 905, 153, 397, 133, 8065, 129, 389, 141, 921

Offset: 0

Ilya Gutkovskiy, Mar 15 2017

nonn

XOR the binary representations of n^2 and (n + 1)^2.

Antti Karttunen, Table of n, a(n) for n = 0..20000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, XOR
Index entries for sequences related to binary expansion of n

Cf. A000290, A003987, A016813 (records), A038712, A112591, A169810.

Cf. also A379007.

Table[BitXor[n^2, (n + 1)^2], {n, 0, 67}]

A283750(n) = bitxor(n^2, (n+1)^2); \\ Indranil Ghosh, Mar 15 2017, Antti Karttunen, Dec 16 2024

def A283750(n): return (n**2)^(n + 1)**2 # Indranil Ghosh, Mar 15 2017

a(n) = A000290(n) XOR A000290(n+1).

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

Original entry on oeis.org

0, 0, 28, 0, 18, 20, 88, 0, 216, 18, 116, 180, 490, 24, 86, 0, 886, 472, 940, 226, 404, 108, 1544, 756, 2028, 74, 500, 200, 1530, 3086, 1120, 0, 3648, 366, 3962, 1160, 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

Offset: 1

Antti Karttunen, Dec 16 2024

nonn

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.

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.

Cf. A000079 (conjectured to give positions of all 0's), A000290, A003987, A033879, A065764, A336701, A378988.

Cf. also A378999, A379007.

Map[BitXor[2*#, DivisorSigma[1, #] + 1] &, Range[100]^2] (* Paolo Xausa, Dec 18 2024 *)

A379009(n) = bitxor(2*(n^2),1+sigma(n^2));

a(n) = A378988(A000290(n)).

cp's OEIS Frontend

A283750 a(n) = n^2 XOR (n + 1)^2.

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