A175524 -id:A175524 - cp's OEIS frontend

A175522 A000120-perfect numbers.

2, 25, 95, 111, 119, 123, 125, 169, 187, 219, 221, 247, 289, 335, 365, 411, 415, 445, 485, 493, 505, 629, 655, 685, 695, 697, 731, 767, 815, 841, 871, 943, 949, 965, 985, 1003, 1139, 1207, 1241, 1261, 1263, 1273, 1343, 1387, 1465, 1469, 1507, 1513, 1529, 1563

Offset: 1

Vladimir Shevelev, Dec 03 2010

nonn

Let A(n), n>=1, be an infinite positive sequence.
We call a number n:
   A-deficient if Sum{d|n, d   A-abundant if Sum{d|n, d A(n),
and
   A-perfect if Sum{d|n, ddepending on the sum over the proper divisors of n.
The definition generalizes the standard nomenclature of deficient (A005100), abundant (A005101) and perfect numbers (A000396), which is recovered by setting A(n) = n = A000027(n).
Conjecture: if there exist infinitely many A-deficient numbers and infinitely many A-abundant numbers, then there exist infinitely many A-perfect numbers.
Note that the sequence contains squares of all Fermat primes larger than 3 (see A019434). [This would also hold for squares of any hypothetical Fermat primes after the fifth one, 65537. Comment clarified by Antti Karttunen, May 14 2015]
A192895(a(n)) = 0. - Reinhard Zumkeller, Jul 12 2011

			Proper divisors of 119 are 1,7,17. Since A000120(1)+A000120(7)+A000120(17)=A000120(119), then 119 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A175524 (deficient version), A175526 (abundant version), A000120, A000396.
Subsequence of A257691 (non-abundant version).
Positions of zeros in A192895.
Cf. A019434, A253204.

import Data.List (elemIndices)
a175522 n = a175522_list !! (n-1)
a175522_list = map (+ 1) $ elemIndices 0 a192895_list
-- Reinhard Zumkeller, Jul 12 2011

binw[n_] := DigitCount[n, 2, 1]; Select[Range[1500], binw[#] == DivisorSum[#, binw[#1] &]/2 &] (* Amiram Eldar, Dec 14 2020 *)

is(n)=sumdiv(n,d,hammingweight(d))==2*hammingweight(n) \\ Charles R Greathouse IV, Jan 28 2016

from sympy import divisors
def A000120(n): return bin(n).count('1')
def aupto(limit):
  alst = []
  for m in range(1, limit+1):
    if A000120(m) == sum(A000120(d) for d in divisors(m)[:-1]): alst += [m]
  return alst
print(aupto(1563)) # Michael S. Branicky, Feb 25 2021

A000120 = lambda x: x.digits(base=2).count(1)
is_A175522 = lambda x: sum(A000120(d) for d in divisors(x)) == 2*A000120(x)
A175522 = filter(is_A175522, IntegerRange(1, 10**4))
# D. S. McNeil, Dec 04 2010

More terms from Amiram Eldar, Feb 18 2019

A175526 A000120-abundant numbers.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 117, 118, 120

Offset: 1

Vladimir Shevelev, Dec 03 2010

nonn

Definition see in A175522. All even numbers > 2 are in the sequence.

A192895(a(n)) > 0. Reinhard Zumkeller, Jul 12 2011

Reinhard Zumkeller (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A175522 (perfect version), A175524 (deficient version), A257691 (complement, non-abundant version).
Cf. A000120, A192895.
Cf. also A005100, A005101.
a(n) differs from A091212(n) and from A205783(n+1) for the first time at n=37, where a(37) = 55, while 55 is missing from both A091212 and A205783.
Differs from A192506 for the first time at n=54, where a(54) = 77, while 77 is missing from A192506.

import Data.List (findIndices)
a175526 n = a175526_list !! (n-1)
a175526_list = map (+ 1) $ findIndices (> 0) a192895_list
-- Reinhard Zumkeller, Jul 12 2011

isA175526 := proc(n) s := 0 ; for d in (numtheory[divisors](n) minus {n}) do s := s+A000120(d) ; end do: evalb(s> A000120(n)) ; end proc:
for n from 1 to 120 do if isA175526(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jul 11 2011

okQ[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[#==n]&]>0; Select[Range[120], okQ] (* Jean-François Alcover, Dec 06 2015 *)

A192895(n) = sumdiv(n, d, hammingweight(d)*(-1)^(d==n)); \\ Charles R Greathouse IV, Feb 07 2013
isA175526(n) = (A192895(n) > 0);
n = 0; i = 0; while(i < 10000, n++; if(isA175526(n), i++; write("b175526.txt", i, " ", n)));
\\ Antti Karttunen, May 11 2015

is(n)=sumdiv(n,d,hammingweight(d))>2*hammingweight(n) \\ Charles R Greathouse IV, Jan 28 2016

is_A175526 = lambda x: sum(A000120(d) for d in divisors(x)) > 2*A000120(x)
A175526 = filter(is_A175526, IntegerRange(1, 10**4))
# D. S. McNeil, Dec 04 2010

A192895 A000120-deficiency of n.

Original entry on oeis.org

-1, 0, -1, 1, -1, 2, -2, 2, 1, 2, -2, 5, -2, 2, 1, 3, -1, 6, -2, 5, 3, 2, -3, 8, 0, 2, 1, 6, -3, 10, -4, 4, 4, 2, 3, 11, -2, 2, 2, 8, -2, 12, -3, 6, 7, 2, -4, 11, 1, 6, 1, 6, -3, 10, 1, 10, 2, 2, -4, 19, -4, 2, 5, 5, 4, 12, -2, 5, 4, 12, -3, 16, -2, 2, 8, 6

Offset: 1

Reinhard Zumkeller, Jul 12 2011

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000120, A033879, A093653, A175522, A175524, A175526, A292257.

Cf. A257691 (positions where a(n) <= 0), A294905 (and its char.fun).

a192895 n =
   sum (map a000120 $ filter ((== 0) . (mod n)) [1..n-1]) - a000120 n
a192895_list = map a192895 [1..]

a[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[# == n]&]; Array[a, 80] (* Jean-François Alcover, Dec 05 2015, adapted from PARI *)

a(n)=sumdiv(n,d,hammingweight(d)*(-1)^(d==n)) \\ Charles R Greathouse IV, Feb 07 2013

from sympy import divisors
def A192895(n): return sum((d.bit_count() if dChai Wah Wu, Jul 25 2023

a(n) = Sum(A000120(d): 1 <= d < n and n mod d = 0) - A000120(n); see A175522 for motivation and more information;
a(A175524(n)) < 0; a(A175522(n)) = 0; a(A175526(n)) > 0.
a(n) = A292257(n) - A000120(n). - Antti Karttunen, Nov 10 2017

A292257 a(n) is the total number of 1's in binary expansion of all proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 4, 1, 7, 1, 5, 5, 4, 1, 8, 1, 7, 6, 5, 1, 10, 3, 5, 5, 9, 1, 14, 1, 5, 6, 4, 6, 13, 1, 5, 6, 10, 1, 15, 1, 9, 11, 6, 1, 13, 4, 9, 5, 9, 1, 14, 6, 13, 6, 6, 1, 23, 1, 7, 11, 6, 6, 14, 1, 7, 7, 15, 1, 18, 1, 5, 12, 9, 7, 16, 1, 13, 9, 5, 1, 24, 5, 6, 7, 13, 1, 26, 7, 11, 8, 7, 6, 16, 1, 11, 10, 15, 1, 14, 1, 13, 18

Offset: 1

Antti Karttunen, Oct 04 2017

If a(n) == A000120(n), then n is in A175522, if a(n) < A000120(n), then n is in A175524, and if a(n) > A000120(n), then n is in A175526.

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

Cf. A000120, A093653, A175522, A175524, A175526, A192895, A290090, A293214.

a[n_] := DivisorSum[n, DigitCount[#, 2, 1] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)
Table[Total[Flatten[IntegerDigits[#,2]&/@Most[Divisors[n]]]],{n,120}] (* Harvey P. Dale, Oct 11 2024 *)

PARI
```
A292257(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA000120(d).
a(n) = A093653(n) - A000120(n).
a(n) = A192895(n) + A000120(n).
a(n) = A001222(A293214(n)).
A000035(a(n)) = A000035(A290090(n)). [Parity-wise equivalent with A290090.]

A257691 Numbers that are not A000120-abundant.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 101, 103, 107, 109, 111, 113, 119, 121, 123, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 211, 219, 221, 223, 227, 229, 233, 239, 241, 247, 251, 257

Offset: 1

Antti Karttunen, May 11 2015

nonn

A000120-nonabundant numbers: Numbers n for which A192895(n) <= 0.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000120, A192895.
Complement of A175526 (A000120-abundant numbers).
Disjoint union of A175522 (A000120-perfect numbers) and A175524 (A000120-deficient numbers).
Differs from A206074(n-1), A186891(n) and A257688(n) for the first time at n=19, where a(19) = 59, while A206074(18) = A186891(19) = A257688(19) = 55, a term missing from here.
Differs from A257689 for the first time at n=24, where a(24) = 79, while A257689(24) = 77, a term missing from here.

A192895[n_] := DivisorSum[n, Total[IntegerDigits[#, 2]]*(-1)^Boole[# == n]& ]; Reap[For[n=1, n<300, n++, If[A192895[n] <= 0, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 05 2015 *)

A192895(n) = sumdiv(n, d, hammingweight(d)*(-1)^(d==n));
isA257691(n) = (A192895(n) <= 0);
n=i=0; while(i < 10000, n++; if(isA257691(n), i++; write("b257691.txt", i, " ", n))); \\ Charles R Greathouse IV, Feb 07 2013

A175548 Binary weight of sigma(n).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 3, 2, 2, 3, 3, 2, 2, 5, 2, 4, 2, 3, 1, 2, 2, 4, 5, 3, 2, 3, 4, 2, 1, 6, 2, 4, 2, 5, 3, 4, 3, 4, 3, 2, 3, 3, 4, 2, 2, 5, 4, 5, 2, 3, 4, 4, 2, 4, 2, 4, 4, 3, 5, 2, 3, 7, 3, 2, 2, 6, 2, 2, 2, 4, 3, 4, 5, 3, 2, 3, 2, 5, 5, 6, 3, 3, 4, 2, 4, 4, 4, 5, 3, 3, 1, 2, 4, 6, 3, 5, 4, 5, 4, 4, 3, 4, 2

Offset: 1

Vladimir Shevelev, Dec 03 2010

The sequence is considered in connection with A175522, A175524, A175526.

a(n)=1 if n is in A046528. - Robert Israel, Nov 07 2017

			a(4) = 3 because the divisors of 4 add up to 7, a number which in binary is written as 3 ones.

Cf. A000203, A000120, A046528, A175522, A175524, A175526.

seq(convert(convert(numtheory:-sigma(n),base,2),`+`),n=1..100); # Robert Israel, Nov 07 2017

Table[Plus@@IntegerDigits[DivisorSigma[1, n], 2], {n, 80}] (* Alonso del Arte, Dec 03 2010 *)

a(n) = hammingweight(sigma(n)); \\ Michel Marcus, Feb 08 2016

a(n) = A000120(A000203(n)).

More terms from Antti Karttunen, Nov 07 2017

A324102 Numbers whose "unary-binary encoded prime factorization" (A156552) is A000120-deficient.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 18, 20, 24, 32, 38, 42, 44, 48, 52, 54, 56, 60, 70, 80, 104, 106, 108, 110, 124, 128, 130, 132, 140, 150, 152, 162, 168, 172, 174, 176, 188, 190, 192, 200, 204, 230, 234, 236, 242, 272, 280, 288, 294, 300, 304, 308, 330, 338, 344, 350, 368, 378, 384, 390, 392, 396, 412, 424, 432, 436, 450, 476, 488, 492, 494, 504, 512

Offset: 1

Antti Karttunen, Feb 18 2019

nonn

Numbers n for which A192895(A156552(n)) < 0.

Numbers n such that A156552(n) is in A175524.

Antti Karttunen, Table of n, a(n) for n = 1..339

Cf. A324101 (complement, apart from 1 which is in neither sequence).

Cf. A175524, A192895.

A178344 a(n) = Sum_i prime(i+1)^b(i) where n = Sum_{i>=0} b(i)*2^e(i) is the binary representation of n.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 15, 16, 17, 18, 19, 20, 21, 22, 21, 22, 23, 24, 25, 26, 27, 28, 18, 19, 20, 21, 22, 23, 24, 25, 24, 25, 26, 27, 28, 29, 30, 31, 28, 29, 30, 31, 32, 33, 34, 35, 34, 35, 36, 37, 38, 39, 40, 41, 23, 24, 25, 26, 27, 28, 29, 30, 29, 30

Offset: 0

Juri-Stepan Gerasimov, May 25 2010, Jan 06 2010

a(0) = 0 might be a more logical value for the initial term. - Antti Karttunen, Sep 28 2018

			a(16)=15 because 10000 is the base-2 representation of n=16 and 11^1 + 7^0 + 5^0 + 3^0 + 2^0 = 15.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000040, A001477, A056791, A007088, A080791, A089625, A143710, A175524.

Cf. A178562 (first differences).

A178344 := proc(n)
    if n = 0 then
        dgs := [0] ;
    else
        dgs := convert(n,base,2) ;
    end if;
    add(ithprime(i)^dgs[i],i=1..nops(dgs)) ;
end proc:
seq(A178344(n),n=0..73) ; # R. J. Mathar, Sep 28 2018

Array[Total@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ IntegerDigits[#, 2]] &, 74, 0] (* Michael De Vlieger, Feb 19 2019 *)

a(n) = my(b=Vecrev(binary(n))); if (n==0, b=[0]); sum(i=1, #b, prime(i)^b[i]); \\ Michel Marcus, Sep 29 2018

For n >= 1, a(n) = A089625(n) + A080791(n). - Antti Karttunen, Sep 28 2018

Offset modified, keyword:base added by R. J. Mathar, May 28 2010

A177052 Ceiling(n/2)-abundant numbers.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228

Offset: 1

Vladimir Shevelev, Dec 09 2010

nonn

For definition, see A175522.

All positive numbers == 0 (mod 6) are in the sequence (basically A008588). In addition, note that all odd primes are ceiling(n/2)-deficient numbers. The first odd term of the sequence is 315.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A177050 (perfect version), A005100, A005101, A175524, A175526, A175811, A175821, A175853.

isok(n) = sumdiv(n, d, (d ceil(n/2); \\ Michel Marcus, Feb 08 2016

is_A177052 = lambda n: sum(ceil(d/2) for d in divisors(n)) > 2*ceil(n/2) # D. S. McNeil, Dec 10 2010

{n : Sum_{d|n, dA004526(1+d) > A004526(1+n)}. [R. J. Mathar, Dec 11 2010]

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A006005 The odd prime numbers together with 1.

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A175522 A000120-perfect numbers.

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A175526 A000120-abundant numbers.

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A192895 A000120-deficiency of n.

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A292257 a(n) is the total number of 1's in binary expansion of all proper divisors of n.

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A257691 Numbers that are not A000120-abundant.

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A175548 Binary weight of sigma(n).

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