A318467 -id:A318467 - cp's OEIS frontend

A378227 XOR-Moebius transform of A318467, where A318467(n) = 2*n XOR sigma(n).

3, 4, 1, 8, 15, 6, 5, 16, 29, 14, 25, 12, 23, 6, 11, 32, 51, 30, 49, 12, 13, 22, 53, 24, 33, 14, 1, 12, 39, 126, 29, 64, 105, 70, 127, 20, 111, 70, 99, 24, 123, 58, 121, 12, 15, 38, 109, 48, 93, 30, 31, 28, 95, 22, 51, 24, 17, 14, 73, 172, 71, 6, 1, 128, 205, 114, 193, 140, 221, 102, 197, 72, 219, 142, 205, 108

Offset: 1

Antti Karttunen, Nov 26 2024

nonn

Unique sequence satisfying SumXOR_{d divides n} a(d) = A318467(n) for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of Xor-Moebius transform.

Cf. A000203, A003987, A256739, A296203, A318467.

Cf. also A083254, A378226.

A318467(n) = bitxor(2*n, sigma(n));
A378227(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A318467(d)))); (v); }

a(n) = 2*A256739(n) XOR A296203(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

A318468 a(n) = 2*n AND A000203(n), where AND is bitwise-and (A004198) and A000203 = sum of divisors.

Original entry on oeis.org

0, 0, 4, 0, 2, 12, 8, 0, 0, 16, 4, 24, 10, 24, 24, 0, 2, 36, 4, 40, 32, 36, 8, 48, 18, 32, 32, 56, 26, 8, 32, 0, 0, 4, 0, 72, 2, 12, 8, 80, 2, 64, 4, 80, 74, 72, 16, 96, 32, 68, 64, 96, 34, 104, 72, 112, 80, 80, 52, 40, 58, 96, 104, 0, 0, 128, 4, 8, 0, 128, 8, 128, 2, 16, 20, 136, 0, 136, 16, 160, 32, 36, 4, 160, 40, 132, 40, 176, 18, 160, 48

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A003987, A318466, A318467.

Cf. also A318458.

Array[BitAnd[2 #, DivisorSigma[1, #]] &, 91] (* Michael De Vlieger, Apr 21 2019 *)

PARI
```
A318468(n) = bitand(2*n,sigma(n));
```

a(n) = A004198(2*n, A000203(n)).

a(n) = A224880(n) - A318466(n) = (A224880(n)-A318467(n))/2.

A318457 a(n) = n XOR A001065(n), where XOR is bitwise-xor (A003987) and A001065 = sum of proper divisors.

Original entry on oeis.org

1, 3, 2, 7, 4, 0, 6, 15, 13, 2, 10, 28, 12, 4, 6, 31, 16, 7, 18, 2, 30, 24, 22, 60, 31, 10, 22, 0, 28, 52, 30, 63, 46, 54, 46, 19, 36, 48, 54, 26, 40, 28, 42, 4, 12, 52, 46, 124, 57, 25, 38, 26, 52, 116, 38, 120, 46, 26, 58, 80, 60, 28, 22, 127, 82, 12, 66, 126, 94, 12, 70, 51, 72, 98, 122, 12, 94, 20, 78, 58, 121, 126, 82, 216, 66

Offset: 1

Antti Karttunen, Aug 26 2018

Cf. A000203, A001065, A003987, A033879, A318456, A318458, A318467.

Cf. A000396 (positions of zeros).

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

PARI
```
A318457(n) = bitxor(n,sigma(n)-n);
```

a(n) = A003987(n, A001065(n)).

a(n) = A000203(n) - 2*A318458(n).

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).

A325635 a(n) = A091255(2n, sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 6, 12, 2, 1, 1, 6, 2, 4, 2, 4, 6, 1, 6, 3, 2, 2, 2, 4, 6, 12, 1, 2, 10, 56, 6, 12, 2, 1, 6, 10, 2, 1, 2, 4, 2, 10, 2, 4, 2, 4, 10, 12, 2, 4, 1, 1, 6, 2, 6, 20, 2, 8, 10, 6, 2, 8, 2, 4, 2, 1, 42, 12, 2, 6, 2, 28, 6, 9, 2, 2, 2, 4, 6, 4, 2, 2, 1, 2, 14, 56, 10, 12, 2, 4, 6, 2, 2, 8, 2, 4, 10, 12, 2, 1, 10, 1, 6, 20, 2, 2, 6

Offset: 1

Antti Karttunen, May 21 2019

nonn

Cf. A000203, A091255, A318467, A325634, A325636, A325638 (n such that a(n) = 2n).

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325635(n) = A091255sq(n+n, sigma(n));

a(n) = A091255(2n, A000203(n)).

a(n) = A091255(2n, A318467(n)) = A091255(A000203(n), A318467(n)).

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

Original entry on oeis.org

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

Antti Karttunen, Dec 16 2024

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.

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.

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

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

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

A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].

Original entry on oeis.org

10, 12, 18, 20, 24, 40, 56, 88, 104, 116, 136, 184, 196, 224, 312, 368, 428, 464, 520, 528, 650, 672, 760, 884, 992, 1472, 1504, 1888, 1952, 2528, 3424, 3724, 4832, 5312, 6464, 7136, 9112, 11096, 11288, 11744, 13216, 15352, 15376, 15872, 15968, 16256, 17816, 17964, 22616, 24448, 26728, 28544, 29296, 30592, 30656

Offset: 1

Antti Karttunen, Jan 05 2025

nonn

Among the first 484 terms, there are no odd numbers, the only squares are 196, 15376, 1032256, and 18 is the only twice square.

			196 is a term as sigma(196) = 399, 2*196 XOR 399 = 7 is not zero, and A048720(7, 89) = 399.

Cf. A000203, A000396, A003987, A048720, A280500, A318467.
Cf. A379234 (subsequence).
Cf. also A097498 (= A153501 U A271816).

divides_in_GF2X(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); !lift(Pa % Pb); };
is_A379236(n) = { my(s=sigma(n), x=bitxor(2*n, s)); (x && divides_in_GF2X(n, x)); };

{k such that k = A048720(A318467(k), x) for some x > 0}.

{k not in A000396 such that A280500(k, A318467(k)) > 0}.

A379234 Numbers k for which k XOR 2*k = sigma(k), where sigma is the sum of divisors function.

Original entry on oeis.org

312, 428, 672, 760, 5009850

Offset: 1

Antti Karttunen, Jan 04 2025

Equally, numbers k such that 2*k XOR sigma(k) = k, i.e., k XOR sigma(k) = 2*k.

If it exists, a(6) > 2^33.

			672 has binary expansion 1010100000_2, and 672 XOR 2*672 has binary expansion 11111100000_2 = 2016 (= 63*32) = sigma(672), so 672 is included in this sequence. Notably, as 672 is also a Fibbinary number (in A003714, no adjacent 1-bits), it follows that 672 XOR 2*672 = 3*672, and thus 672 is also a 3-perfect number, A005820.

Cf. A000203, A003714, A003987, A005820, A048724.
Fixed points of A318467.
Subsequence of A379236.

is_A379234(n) = (bitxor(2*n,n)==sigma(n));

{k such that A000203(k) = A048724(k)}.

cp's OEIS Frontend

A378227 XOR-Moebius transform of A318467, where A318467(n) = 2*n XOR sigma(n).

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A156552 Unary-encoded compressed factorization of natural numbers.

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A318468 a(n) = 2*n AND A000203(n), where AND is bitwise-and (A004198) and A000203 = sum of divisors.

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A318457 a(n) = n XOR A001065(n), where XOR is bitwise-xor (A003987) and A001065 = sum of proper divisors.

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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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A325635 a(n) = A091255(2n, sigma(n)).

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

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A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].

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