A326133 -id:A326133 - 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

A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 15370304, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056

Offset: 1

Antti Karttunen, Jun 09 2019

No further terms below 2^31.
See also comments in A326133.
The quotients A000120(k)/(sigma(k)-A005187(k)) for these terms are: 1, 1, 5, 1, 3, 1, 5, 9, 2, 2, 1, 2, 2. Ones occur at the positions of perfect numbers.
a(18) > 10^11. - Amiram Eldar, Jan 03 2021

			110 is "1101110" in binary, thus A000120(110) = 5. Sigma(110) = 216, while A005187(110) = 215, thus as 5 = 5*(216-215), 110 is included in this sequence.

Intersection of A326132 and A326133, also of A326132 and A326138.
Cf. A000120, A000396 (subsequence), A005187, A294898, A295296, A295297, A295298, A295299, A326130.
Cf. also A325981, A326141.

q[n_] := Module[{bw = DigitCount[n, 2, 1], ab = DivisorSigma[1, n] - 2*n, sum}, (sum = ab + bw) > 0 && Divisible[bw, sum]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 03 2021 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326131(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == t); };

a(14)-a(17) from Amiram Eldar, Jan 03 2021

A326132 Numbers n for which A294898(n) is not zero and A294898(n) divides A000120(n); numbers for which A326130(n) = abs(A294898(n)).

Original entry on oeis.org

5, 6, 7, 14, 15, 28, 44, 52, 110, 152, 184, 496, 592, 884, 1012, 1155, 2144, 2272, 8128, 8384, 12008, 18632, 18904, 33664, 63248, 70564, 85936, 100804, 116624, 318250, 527872, 1090912, 1360810, 1503370, 1788490, 2085710, 2102272, 3477608, 4495808, 8394752, 15370304, 16102808, 26347688, 29322008, 33550336, 73995392

Offset: 1

Antti Karttunen, Jun 11 2019

nonn

Cf. A000360, A326131 (subsequences), A326133.

Cf. A000120, A000203, A005187, A294898, A295296.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326132(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == abs(t)); };

A326138 Numbers k such that A005187(k) < sigma(k) <= 2k, where A005187(k) = 2k - {binary weight of k}.

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 391612, 15370304, 17619844, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056, 42783299288, 80999455688, 137438691328, 217898810368, 546409576448, 1081071376208, 1661355408388

Offset: 1

Antti Karttunen, Jun 13 2019

nonn

Non-abundant numbers whose deficiency (A033879) is less than their binary weight (A000120).

No other terms below < 2^31.

			815634435 = 3*5*7*11*547*1291 is included as in base-2 (A007088) it is written as 110000100111011001100000000011_2, thus A000120(815634435) = 12, while its nonnegative deficiency (A033879) is 2*815634435 - sigma(815634435) = 6 < 12.

Cf. A000120, A000203, A000396 (subsequence), A005187, A033879, A294898, A295296 (deficiency equals binary weight), A326131, A326132.
Intersection of A263837 and A326133.
Cf. also A087485, A141548, A188597.

isA326138(n) = { my(s=sigma(n)); (s>A005187(n)&&s<=(n+n)); };

a(16)-a(26) from Giovanni Resta, Jun 16 2019

cp's OEIS Frontend

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

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A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

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A326132 Numbers n for which A294898(n) is not zero and A294898(n) divides A000120(n); numbers for which A326130(n) = abs(A294898(n)).

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A326138 Numbers k such that A005187(k) < sigma(k) <= 2k, where A005187(k) = 2k - {binary weight of k}.

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