A378988 a(n) = 2*n XOR 1+sigma(n), where XOR is bitwise-xor, A003987.

0, 0, 3, 0, 13, 1, 7, 0, 28, 7, 27, 5, 21, 5, 7, 0, 49, 12, 51, 3, 11, 9, 55, 13, 18, 31, 31, 1, 37, 117, 31, 0, 115, 115, 119, 20, 109, 113, 119, 11, 121, 53, 123, 13, 21, 21, 111, 29, 88, 58, 47, 11, 93, 21, 39, 9, 35, 47, 75, 209, 69, 29, 23, 0, 215, 21, 195, 247, 235, 29, 199, 84, 217, 231, 235, 21, 251, 53, 207

Offset: 1

Author

Antti Karttunen, Dec 16 2024

Keywords

• nonn
• look

Comments

For any hypothetical quasiperfect number q (for which sigma(q) = 2*q+1, see A336701), a(q) would be equal to 2*q XOR 2*q+2 = 2*(q XOR q+1) = 2*A038712(1+q) = A100892(1+q).

See also A000079 and A235796 concerning the "almost perfect" or "least deficient" numbers that give positions of 0's here.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..32768
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000079 (conjectured to give positions of all 0's), A000396 (positions of 1's), A000203, A003987, A028982 (positions of even terms), A028983 (of odd terms), A038712, A100892, A318467, A336701, A378998, A379009 [= a(n^2)].

Cf. also A235796, A294898, A294899, A318457.

Programs

Mathematica

Array[BitXor[2*#, DivisorSigma[1, #] + 1] &, 100] (* Paolo Xausa, Dec 16 2024 *)

PARI

A378988(n) = bitxor(n+n,1+sigma(n));

Formula

For all n in A028983, a(n) = 2n+1 XOR sigma(n) = 1+A318467(n).