A378988 -id:A378988 - cp's OEIS frontend

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 7, -6, 9, 1, 2, 0, 14, -5, 15, -4, 7, 5, 18, -14, 16, 7, 10, -3, 24, -16, 25, 0, 16, 12, 19, -21, 33, 13, 18, -12, 37, -15, 38, 1, 8, 16, 41, -30, 38, 4, 26, 3, 48, -16, 33, -11, 30, 22, 53, -52, 55, 23, 16, 0, 44, -14, 63, 8, 39, -7, 66, -53, 69, 31, 22, 9, 54, -16, 73, -28, 38, 35, 78, -59, 58

Offset: 1

Antti Karttunen, Nov 25 2017

sign

"Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - Antti Karttunen, Dec 16 2024

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.

Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

Array[IntegerExponent[(2 #)!, 2] - DivisorSigma[1, #] &, 85] (* Michael De Vlieger, Nov 26 2017 *)

A294898(n) = ((2*n-sigma(n))-hammingweight(n)); \\ Antti Karttunen, Dec 16 2024

(define (A294898 n) (- (A005187 n) (A000203 n)))

a(n) = A005187(n) - A000203(n).
a(n) = A011371(n) - A001065(n).
a(n) = A033879(n) - A000120(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - Amiram Eldar, Feb 22 2024

Name edited by Antti Karttunen, Dec 16 2024

A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

Original entry on oeis.org

3, 7, 2, 15, 12, 0, 6, 31, 31, 6, 26, 4, 20, 4, 6, 63, 48, 3, 50, 2, 10, 8, 54, 12, 45, 30, 30, 0, 36, 116, 30, 127, 114, 114, 118, 19, 108, 112, 118, 10, 120, 52, 122, 12, 20, 20, 110, 28, 91, 57, 46, 10, 92, 20, 38, 8, 34, 46, 74, 208, 68, 28, 22, 255, 214, 20, 194, 246, 234, 28, 198, 83, 216, 230, 234, 20, 250, 52, 206, 26

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003987, A318457, A318458, A318466, A318468.
Cf. A000396 (positions of zeros), A378227 (XOR-Moebius transform), A379234 (fixed points), A379236.
Cf. also A294899, A318457, A378988.

Table[BitXor[2n,DivisorSigma[1,n]],{n,80}] (* Harvey P. Dale, Oct 30 2022 *)

PARI
```
A318467(n) = bitxor(2*n,sigma(n));
```

a(n) = A003987(2*n, A000203(n)).
a(n) = A224880(n) - 2*A318468(n).
a(n) = 2*n XOR (A318457(n)+2*A318458(n)). - Antti Karttunen, Jan 08 2025

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

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

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A318467 a(n) = 2*n XOR A000203(n), where XOR is bitwise-xor (A003987) and A000203 = sum of divisors.

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

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