A318468 -id:A318468 - cp's OEIS frontend

A324652 Numbers k such that A318468(k) (bitwise-AND of 2k and sigma(k)) is equal to 2k.

6, 12, 18, 20, 24, 28, 36, 40, 48, 56, 80, 88, 96, 100, 104, 112, 160, 176, 192, 196, 200, 204, 208, 220, 224, 260, 264, 272, 304, 320, 336, 352, 368, 384, 392, 416, 448, 464, 496, 544, 550, 580, 608, 640, 648, 650, 672, 704, 736, 768, 784, 832, 896, 928, 992, 1032, 1040, 1044, 1056, 1060, 1068, 1088, 1104, 1120, 1184, 1216

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Positions of zeros in A324658, fixed points of A324659.
Intersection with A324649 gives A324643.
Intersection with A324726 gives A000396.
In the range 1..2^32 there are only 22 odd terms. See A324647.

Cf. A000203, A191218, A318468, A324647, A324649, A324658, A324659, A324722, A324726.

Some subsequences: A000396, A324643, A324647 (the odd terms).

Select[Range[2000], 2*# == BitAnd[2*#, DivisorSigma[1, #]] &] (* Paolo Xausa, Mar 11 2024 *)

for(n=1,oo,if((n+n)==bitand(2*n,sigma(n)), print1(n, ", ")))

A324659 a(n) = (1/2)A318468(n), where A318468(n) is bitwise-AND of 2n and sigma(n).

Original entry on oeis.org

0, 0, 2, 0, 1, 6, 4, 0, 0, 8, 2, 12, 5, 12, 12, 0, 1, 18, 2, 20, 16, 18, 4, 24, 9, 16, 16, 28, 13, 4, 16, 0, 0, 2, 0, 36, 1, 6, 4, 40, 1, 32, 2, 40, 37, 36, 8, 48, 16, 34, 32, 48, 17, 52, 36, 56, 40, 40, 26, 20, 29, 48, 52, 0, 0, 64, 2, 4, 0, 64, 4, 64, 1, 8, 10, 68, 0, 68, 8, 80, 16, 18, 2, 80, 20, 66, 20, 88, 9, 80

Offset: 1

Antti Karttunen, Mar 14 2019

Cf. A000203, A004198, A318458, A318468, A324648, A324658.

Cf. A324652 (fixed points).

Array[BitAnd[2*#, DivisorSigma[1, #]]/2 &, 100] (* Paolo Xausa, Mar 11 2024 *)

PARI
```
A324659(n) = (bitand(2*n,sigma(n))/2);
```

a(n) = A318468(n)/2.

a(n) = n - A324658(n).

A156552 Unary-encoded compressed factorization of natural numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 18, 33, 256, 23, 12, 65, 14, 35, 512, 21, 1024, 31, 34, 129, 20, 27, 2048, 257, 66, 39, 4096, 37, 8192, 67, 22, 513, 16384, 47, 24, 25, 130, 131, 32768, 29, 36, 71, 258, 1025, 65536, 43, 131072, 2049, 38, 63, 68, 69, 262144

Offset: 1

Leonid Broukhis, Feb 09 2009

The primes become the powers of 2 (2 -> 1, 3 -> 2, 5 -> 4, 7 -> 8); the composite numbers are formed by taking the values for the factors in the increasing order, multiplying them by the consecutive powers of 2, and summing. See the Example section.
From Antti Karttunen, Jun 27 2014: (Start)
The odd bisection (containing even terms) halved gives A244153.
The even bisection (containing odd terms), when one is subtracted from each and halved, gives this sequence back.
(End)
Question: Are there any other solutions that would satisfy the recurrence r(1) = 0; and for n > 1, r(n) = Sum_{d|n, d>1} 2^A033265(r(d)), apart from simple variants 2^k * A156552(n)? See also A297112, A297113. - Antti Karttunen, Dec 30 2017

			For 84 = 2*2*3*7 -> 1*1 + 1*2 + 2*4 + 8*8 =  75.
For 105 = 3*5*7 -> 2*1 + 4*2 + 8*4 = 42.
For 137 = p_33 -> 2^32 = 4294967296.
For 420 = 2*2*3*5*7 -> 1*1 + 1*2 + 2*4 + 4*8 + 8*16 = 171.
For 147 = 3*7*7 = p_2 * p_4 * p_4 -> 2*1 + 8*2 + 8*4 = 50.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1024 terms from Antti Karttunen)
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]
A puzzle by Sergey Orlov (in Russian)
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization

One less than A005941.
Inverse permutation: A005940 with starting offset 0 instead of 1.
Cf. A000079, A000120, A001222, A052126, A054429, A061395, A064216, A064989, A003188, A243071, A243065-A243066, A244153, A243354, A112798, A125106, A056239, A161511.
Cf. also A297106, A297112 (Möbius transform), A297113, A153013, A290308, A300827, A323243, A323244, A323247, A324201, A324812 (n for which a(n) is a square), A324813, A324822, A324823, A324398, A324713, A324815, A324819, A324865, A324866, A324867.
Other related permutations: A253551, A253792, A253564, A253791, A277195, A297163, A297164, A297165, A297166, A302023, A305418, A322863, A322864.

Table[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[ Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ n]], {n, 67}] (* Michael De Vlieger, Sep 08 2016 *)

a(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ David A. Corneth, Mar 08 2019

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n)))); \\ (based on the given recurrence) - Antti Karttunen, Mar 08 2019

# Program corrected per instructions from Leonid Broukhis. - Antti Karttunen, Jun 26 2014
# However, it gives correct answers only up to n=136, before corruption by a wrap-around effect.
# Note that the correct answer for n=137 is A156552(137) = 4294967296.
$max = $ARGV[0];
$pow = 0;
foreach $i (2..$max) {
@a = split(/ /, `factor $i`);
shift @a;
$shift = 0;
$cur = 0;
while ($n = int shift @a) {
$prime{$n} = 1 << $pow++ if !defined($prime{$n});
$cur |= $prime{$n} << $shift++;
}
print "$cur, ";
}
print "\n";
(Scheme, with memoization-macro definec from Antti Karttunen's IntSeq-library, two different implementations)
(definec (A156552 n) (cond ((= n 1) 0) (else (+ (A000079 (+ -2 (A001222 n) (A061395 n))) (A156552 (A052126 n))))))
(definec (A156552 n) (cond ((= 1 n) (- n 1)) ((even? n) (+ 1 (* 2 (A156552 (/ n 2))))) (else (* 2 (A156552 (A064989 n))))))
;; Antti Karttunen, Jun 26 2014

from sympy import primepi, factorint
def A156552(n): return sum((1<Chai Wah Wu, Mar 10 2023

From Antti Karttunen, Jun 26 2014: (Start)
a(1) = 0, a(n) = A000079(A001222(n)+A061395(n)-2) + a(A052126(n)).
a(1) = 0, a(2n) = 1+2*a(n), a(2n+1) = 2*a(A064989(2n+1)). [Compare to the entanglement recurrence A243071].
For n >= 0, a(2n+1) = 2*A244153(n+1). [Follows from the latter clause of the above formula.]
a(n) = A005941(n) - 1.
As a composition of related permutations:
a(n) = A003188(A243354(n)).
a(n) = A054429(A243071(n)).
For all n >= 1, A005940(1+a(n)) = n and for all n >= 0, a(A005940(n+1)) = n. [The offset-0 version of A005940 works as an inverse for this permutation.]
This permutations also maps between the partition-lists A112798 and A125106:
A056239(n) = A161511(a(n)). [The sums of parts of each partition (the total sizes).]
A003963(n) = A243499(a(n)). [And also the products of those parts.]
(End)
From Antti Karttunen, Oct 09 2016: (Start)
A161511(a(n)) = A056239(n).
A029837(1+a(n)) = A252464(n). [Binary width of terms.]
A080791(a(n)) = A252735(n). [Number of nonleading 0-bits.]
A000120(a(n)) = A001222(n). [Binary weight.]
For all n >= 2, A001511(a(n)) = A055396(n).
For all n >= 2, A000120(a(n))-1 = A252736(n). [Binary weight minus one.]
A252750(a(n)) = A252748(n).
a(A250246(n)) = A252754(n).
a(A005117(n)) = A277010(n). [Maps squarefree numbers to a permutation of A003714, fibbinary numbers.]
A085357(a(n)) = A008966(n). [Ditto for their characteristic functions.]
For all n >= 0:
a(A276076(n)) = A277012(n).
a(A276086(n)) = A277022(n).
a(A260443(n)) = A277020(n).
(End)
From Antti Karttunen, Dec 30 2017: (Start)
For n > 1, a(n) = Sum_{d|n, d>1} 2^A033265(a(d)). [See comments.]
More linking formulas:
A106737(a(n)) = A000005(n).
A290077(a(n)) = A000010(n).
A069010(a(n)) = A001221(n).
A136277(a(n)) = A181591(n).
A132971(a(n)) = A008683(n).
A106400(a(n)) = A008836(n).
A268411(a(n)) = A092248(n).
A037011(a(n)) = A010052(n) [conjectured, depends on the exact definition of A037011].
A278161(a(n)) = A046951(n).
A001316(a(n)) = A061142(n).
A277561(a(n)) = A034444(n).
A286575(a(n)) = A037445(n).
A246029(a(n)) = A181819(n).
A278159(a(n)) = A124859(n).
A246660(a(n)) = A112624(n).
A246596(a(n)) = A069739(n).
A295896(a(n)) = A053866(n).
A295875(a(n)) = A295297(n).
A284569(a(n)) = A072411(n).
A286574(a(n)) = A064547(n).
A048735(a(n)) = A292380(n).
A292272(a(n)) = A292382(n).
A244154(a(n)) = A048673(n), a(A064216(n)) = A244153(n).
A279344(a(n)) = A279339(n), a(A279338(n)) = A279343(n).
a(A277324(n)) = A277189(n).
A037800(a(n)) = A297155(n).
For n > 1, A033265(a(n)) = 1+A297113(n).
(End)
From Antti Karttunen, Mar 08 2019: (Start)
a(n) = A048675(n) + A323905(n).
a(A324201(n)) = A000396(n), provided there are no odd perfect numbers.
The following sequences are derived from or related to the base-2 expansion of a(n):
A000265(a(n)) = A322993(n).
A002487(a(n)) = A323902(n).
A005187(a(n)) = A323247(n).
A324288(a(n)) = A324116(n).
A323505(a(n)) = A323508(n).
A079559(a(n)) = A323512(n).
A085405(a(n)) = A323239(n).
The following sequences are obtained by applying to a(n) a function that depends on the prime factorization of its argument, which goes "against the grain" because a(n) is the binary code of the factorization of n, which in these cases is then factored again:
A000203(a(n)) = A323243(n).
A033879(a(n)) = A323244(n) = 2*a(n) - A323243(n),
A294898(a(n)) = A323248(n).
A000005(a(n)) = A324105(n).
A000010(a(n)) = A324104(n).
A083254(a(n)) = A324103(n).
A001227(a(n)) = A324117(n).
A000593(a(n)) = A324118(n).
A001221(a(n)) = A324119(n).
A009194(a(n)) = A324396(n).
A318458(a(n)) = A324398(n).
A192895(a(n)) = A324100(n).
A106315(a(n)) = A324051(n).
A010052(a(n)) = A324822(n).
A053866(a(n)) = A324823(n).
A001065(a(n)) = A324865(n) = A323243(n) - a(n),
A318456(a(n)) = A324866(n) = A324865(n) OR a(n),
A318457(a(n)) = A324867(n) = A324865(n) XOR a(n),
A318458(a(n)) = A324398(n) = A324865(n) AND a(n),
A318466(a(n)) = A324819(n) = A323243(n) OR 2*a(n),
A318467(a(n)) = A324713(n) = A323243(n) XOR 2*a(n),
A318468(a(n)) = A324815(n) = A323243(n) AND 2*a(n).
(End)

More terms from Antti Karttunen, Jun 28 2014

A318458 a(n) = n AND A001065(n), where AND is bitwise-and (A004198) & A001065 = sum of proper divisors.

Original entry on oeis.org

0, 0, 1, 0, 1, 6, 1, 0, 0, 8, 1, 0, 1, 10, 9, 0, 1, 16, 1, 20, 1, 6, 1, 0, 0, 16, 9, 28, 1, 10, 1, 0, 1, 0, 1, 36, 1, 6, 1, 32, 1, 34, 1, 40, 33, 10, 1, 0, 0, 34, 17, 36, 1, 2, 17, 0, 17, 32, 1, 44, 1, 34, 41, 0, 1, 66, 1, 0, 1, 66, 1, 72, 1, 8, 1, 64, 1, 74, 1, 64, 0, 0, 1, 4, 21, 6, 1, 88, 1, 16, 17, 76, 1, 18, 25, 0, 1, 64, 33, 100, 1, 98, 1, 104, 65

Offset: 1

Antti Karttunen, Aug 26 2018

The peculiar look of the scatterplot is partly an artifact of the logarithmic scale. Compare also to the scatterplot of A318468.

Cf. A000203, A001065, A004198, A033879, A318456, A318457.

Cf. also A318468, A318508, A318518, A318463.

[SumOfDivisors(n)-BitwiseOr(n, SumOfDivisors(n)-n): n in [1..100]]; // Vincenzo Librandi, Aug 29 2018

Table[BitAnd[n, DivisorSigma[1, n] - n], {n, 100}] (* Vincenzo Librandi, Aug 29 2018 *)

PARI
```
A318458(n) = bitand(n,sigma(n)-n);
```

a(n) = A004198(n, A001065(n)).

a(n) = A000203(n) - A318456(n) = (A000203(n)-A318457(n))/2.

A324647 Odd numbers k such that 2k is equal to bitwise-AND of 2k and sigma(k).

Original entry on oeis.org

1116225, 1245825, 1380825, 2127825, 10046025, 16813125, 203753025, 252880425, 408553425, 415433025, 740361825, 969523425, 1369580625, 1612924425, 1763305425, 2018027025, 2048985225, 2286684225, 3341556225, 3915517725, 3985769025, 4051698525, 7085469825, 7520472225

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

If this sequence has no terms common with A324649 (A324897, A324898), or no terms common with A324727, then there are no odd perfect numbers.
First 22 terms factored:
     1116225 = 3^2 * 5^2 * 11^2 * 41
     1245825 = 3^2 * 5^2 *  7^2 * 113
     1380825 = 3^2 * 5^2 * 19^2 * 17    [Here the unitary prime is not the largest]
     2127825 = 3^2 * 5^2 *  7^2 * 193
    10046025 = 3^4 * 5^2 * 11^2 * 41
    16813125 = 3^2 * 5^4 *  7^2 * 61
   203753025 = 3^2 * 5^2 *  7^2 * 18481
   252880425 = 3^2 * 5^2 *  7^2 * 22937
   408553425 = 3^2 * 5^2 *  7^2 * 37057
   415433025 = 3^2 * 5^2 *  7^4 * 769
   740361825 = 3^2 * 5^2 *  7^2 * 67153
   969523425 = 3^4 * 5^2 * 13^2 * 2833
  1369580625 = 3^2 * 5^4 *  7^2 * 4969
  1612924425 = 3^2 * 5^2 *  7^2 * 146297
  1763305425 = 3^2 * 5^2 *  7^2 * 159937
  2018027025 = 3^2 * 5^2 *  7^2 * 183041
  2048985225 = 3^2 * 5^2 *  7^2 * 185849
  2286684225 = 3^2 * 5^2 *  7^2 * 207409
  3341556225 = 3^2 * 5^2 *  7^2 * 303089
  3915517725 = 3^4 * 5^2 *  7^2 * 39461
  3985769025 = 3^4 * 5^2 *  7^2 * 40169
  4051698525 = 3^2 * 5^2 *  7^2 * 367501.
Compare the above factorizations to the various constraints listed for odd perfect numbers in the Wikipedia article. However, this is NOT a subsequence of A191218 (A228058), see below.
The first terms that do not belong to A191218 are 399736269009 = (3 * 7^2 * 11 * 17 * 23)^2 and 1013616036225 = (3^2 * 5 * 13 * 1721)^2, that both occur instead in A325311. The first terms with omega(n) <> 4 are 9315603297, 60452246925, 68923392525, and 112206463425. They factor as 3^2 * 7^2 * 11^2 * 13^2 * 1033, 3^2 * 5^2 * 7^2 * 17^2 * 18973, 3^2 * 5^2 * 13^2 * 19^2 * 5021, 3^2 * 5^2 * 7^2 * 199^2 * 257. - Giovanni Resta, Apr 21 2019
From Antti Karttunen, Jan 13 2025: (Start)
Because of the "monotonic property" of bitwise-and, this is a subsequence of nondeficient numbers (A023196).
Both odd perfect numbers, and quasiperfect numbers, if such numbers exist at all, would satisfy the condition for being included in this sequence. Furthermore, any term must be either an odd square with an odd abundancy (in A156942), which subset is given in A379490 (where quasiperfect numbers must thus reside, if they exist), or be included in A228058, i.e., satisfy the Euler's criteria for odd perfect numbers.
(End)

Giovanni Resta, Table of n, a(n) for n = 1..500
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Odd terms of A324652.

Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311, A379490 (square terms).

for(n=1,oo,if((n%2)&&((2*n)==bitand(2*n,sigma(n))),print1(n,", ")));

{Odd k such that 2k = A318468(k)}.

a(23)-a(24) from Giovanni Resta, Apr 21 2019

A318466 a(n) = 2*n OR A000203(n), where OR is bitwise-or (A003986) and A000203 = sum of divisors.

Original entry on oeis.org

3, 7, 6, 15, 14, 12, 14, 31, 31, 22, 30, 28, 30, 28, 30, 63, 50, 39, 54, 42, 42, 44, 62, 60, 63, 62, 62, 56, 62, 124, 62, 127, 114, 118, 118, 91, 110, 124, 126, 90, 122, 116, 126, 92, 94, 92, 126, 124, 123, 125, 110, 106, 126, 124, 110, 120, 114, 126, 126, 248, 126, 124, 126, 255, 214, 148, 198, 254, 234, 156, 206

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003986, A224880, A318467, A318468.

Cf. also A318456.

Array[BitOr[2 #, DivisorSigma[1, #]] &, 71] (* Michael De Vlieger, Mar 30 2019 *)

PARI
```
A318466(n) = bitor(2*n,sigma(n));
```

from sympy import divisor_sigma
def A318466(n): return (n<<1)|int(divisor_sigma(n)) # Chai Wah Wu, Jul 10 2022

a(n) = A003986(2*n, A000203(n)).

a(n) = A224880(n) - A318468(n).

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

A324718 Odd numbers n for which bitand(2n,sigma(n)) = 2*bitand(n,sigma(n)-n), where bitand is bitwise-AND, A004198.

Original entry on oeis.org

1, 5, 9, 17, 37, 41, 73, 137, 149, 153, 257, 261, 277, 293, 337, 405, 521, 529, 549, 577, 593, 641, 661, 673, 677, 1025, 1033, 1061, 1093, 1097, 1109, 1153, 1193, 1289, 1297, 1301, 1321, 1361, 2053, 2069, 2081, 2089, 2097, 2113, 2129, 2209, 2213, 2225, 2309, 2341, 2377, 2389, 2593, 2633, 2689, 2693, 2729, 2825, 4129, 4133, 4177, 4229

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Odd numbers n for which 2*A318458(n) = A318468(n). If there are no common terms with A324719, then there are no odd perfect numbers.

This is not a subsequence of A191218, because terms 1, 9, 529, 2209, 10609, 77841, 83521, 263169, 279841, 330625, 528529, ... are not present in A191218.

Subsequence of A324639.

Cf. A004198, A191218, A318458, A318468, A324647, A324719, A324727.

Select[Range[1, 10^4, 2], Block[{s = DivisorSigma[1, #]}, BitAnd[2*#, s] == 2* BitAnd[#, s-#]] &] (* Paolo Xausa, Mar 11 2024 *)

for(n=1,oo,if((n%2) && (bitand(2*n,sigma(n)) == 2*bitand(n,sigma(n)-n)),print1(n, ", ")));

A324815 a(n) = 2*A156552(n) AND A323243(n), where AND is bitwise-and, A004198.

Original entry on oeis.org

0, 0, 0, 4, 0, 2, 0, 8, 12, 0, 0, 4, 0, 2, 16, 24, 0, 10, 0, 4, 36, 0, 0, 8, 24, 0, 24, 0, 0, 32, 0, 32, 4, 0, 40, 32, 0, 2, 128, 8, 0, 2, 0, 4, 36, 0, 0, 16, 48, 18, 4, 4, 0, 26, 72, 8, 512, 2, 0, 4, 0, 0, 12, 104, 8, 0, 0, 0, 4, 2, 0, 72, 0, 0, 32, 0, 80, 0, 0, 16, 8, 0, 0, 20, 256, 0, 2048, 0, 0, 74, 128, 0, 0, 0, 520, 56, 0, 32, 128, 64, 0, 2, 0, 8, 64

Offset: 1

Antti Karttunen, Mar 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003987, A004198, A139256, A156552, A318468, A323243, A324201, A324716, A324816.

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth
A324815(n) = bitand(2*A156552(n),A323243(n)); \\ Needs code also from A323243.

a(n) = 2*A156552(n) AND A323243(n), where AND is A004198.
a(n) = 2*A156552(n) - A324716(n) = 2*A156552(n) XOR A324716(n), where XOR is A003987.
For n > 1, a(n) = A318468(A156552(n)).
a(p) = 0 for all primes p.
a(A324201(n)) = A139256(n).
A000120(a(n)) = A324816(n).

A324658 a(n) = n - A324659(n), where A324659(n) is half of bitwise-AND of 2*n and sigma(n).

Original entry on oeis.org

1, 2, 1, 4, 4, 0, 3, 8, 9, 2, 9, 0, 8, 2, 3, 16, 16, 0, 17, 0, 5, 4, 19, 0, 16, 10, 11, 0, 16, 26, 15, 32, 33, 32, 35, 0, 36, 32, 35, 0, 40, 10, 41, 4, 8, 10, 39, 0, 33, 16, 19, 4, 36, 2, 19, 0, 17, 18, 33, 40, 32, 14, 11, 64, 65, 2, 65, 64, 69, 6, 67, 8, 72, 66, 65, 8, 77, 10, 71, 0, 65, 64, 81, 4, 65, 20, 67, 0, 80

Offset: 1

Antti Karttunen, Mar 14 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A000203, A004198, A318468, A324648, A324659.

Cf. A324652 (positions of zeros).

Array[# - BitAnd[2*#, DivisorSigma[1, #]]/2 &, 100] (* Paolo Xausa, Mar 13 2024 *)

A324658(n) = (n-(bitand(2*n,sigma(n))/2));

a(n) = n - A324659(n) = n - A318468(n)/2 = n - ((2*n AND sigma(n))/2).

cp's OEIS Frontend

A324652 Numbers k such that A318468(k) (bitwise-AND of 2*k and sigma(k)) is equal to 2*k.

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A324659 a(n) = (1/2)*A318468(n), where A318468(n) is bitwise-AND of 2*n and sigma(n).

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A156552 Unary-encoded compressed factorization of natural numbers.

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A318458 a(n) = n AND A001065(n), where AND is bitwise-and (A004198) & A001065 = sum of proper divisors.

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A324647 Odd numbers k such that 2*k is equal to bitwise-AND of 2*k and sigma(k).

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A318466 a(n) = 2*n OR A000203(n), where OR is bitwise-or (A003986) and A000203 = sum of divisors.

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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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A324652 Numbers k such that A318468(k) (bitwise-AND of 2k and sigma(k)) is equal to 2k.

A324659 a(n) = (1/2)A318468(n), where A318468(n) is bitwise-AND of 2n and sigma(n).

A324647 Odd numbers k such that 2k is equal to bitwise-AND of 2k and sigma(k).