A005187 -id:A005187 - cp's OEIS frontend

A279357 a(n) = A005187(n) XOR A005187(n+1).

1, 2, 7, 3, 15, 2, 1, 4, 31, 2, 1, 5, 1, 14, 3, 5, 63, 2, 1, 5, 1, 14, 3, 4, 1, 30, 3, 7, 3, 14, 1, 6, 127, 2, 1, 5, 1, 14, 3, 4, 1, 30, 3, 7, 3, 14, 1, 7, 1, 62, 3, 7, 3, 14, 1, 4, 3, 30, 1, 5, 1, 2, 15, 7, 255, 2, 1, 5, 1, 14, 3, 4, 1, 30, 3, 7, 3, 14, 1, 7, 1, 62, 3, 7, 3, 14, 1, 4, 3, 30, 1, 5, 1, 2, 15, 6, 1, 126, 3, 7, 3, 14, 1, 4, 3, 30

Offset: 0

Antti Karttunen, Mar 15 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A001511, A003987, A005187, A279353.

Function[t, BitXor @@ # & /@ Transpose@ {Most@ t, Rest@ t}]@ Table[2 n - DigitCount[2 n, 2, 1], {n, 0, 106}] (* Michael De Vlieger, Mar 15 2017, after Harvey P. Dale at A005187 *)

(define (A279357 n) (A003987bi (A005187 (+ 1 n)) (A005187 n))) ;; Here A003987bi implements a bitwise-XOR (A003987).

a(n) = A005187(n) XOR A005187(n+1).

A283997 a(n) = n XOR A005187(floor(n/2)), where XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 2, 3, 15, 14, 2, 3, 6, 7, 5, 4, 31, 30, 2, 3, 6, 7, 5, 4, 14, 15, 13, 12, 5, 4, 4, 5, 63, 62, 2, 3, 6, 7, 5, 4, 14, 15, 13, 12, 5, 4, 4, 5, 30, 31, 29, 28, 5, 4, 4, 5, 13, 12, 12, 13, 4, 5, 7, 6, 127, 126, 2, 3, 6, 7, 5, 4, 14, 15, 13, 12, 5, 4, 4, 5, 30, 31, 29, 28, 5, 4, 4, 5, 13, 12, 12, 13, 4, 5, 7, 6, 62, 63, 61, 60, 5, 4, 4, 5, 13, 12, 12

Offset: 0

Antti Karttunen, Mar 19 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A003986, A005187, A006068, A283996, A283998, A283999.

Cf. also A279357, A283477.

Table[BitXor[n, 2 # - DigitCount[2 #, 2, 1] &@ Floor[n/2]], {n, 0, 106}] (* Michael De Vlieger, Mar 20 2017 *)

b(n) = if(n<1, 0, b(n\2) + n%2);
A(n) = 2*n - b(2*n);
for(n=0, 110, print1(bitxor(n, A(floor(n/2))),", ")) \\ Indranil Ghosh, Mar 25 2017

def A(n): return 2*n - bin(2*n)[2:].count("1")
print([n^A(n//2) for n in range(111)]) # Indranil Ghosh, Mar 25 2017

(define (A283997 n) (A003987bi n (A005187 (floor->exact (/ n 2))))) ;; Where A003987bi implements bitwise-XOR (A003987).

a(n) = n XOR A005187(floor(n/2)), where XOR is bitwise-xor (A003987).
a(n) = A283996(n) - A283998(n).
a(n) = A005187(n) - 2*A283998(n).
a(n) = A006068(n) XOR A283999(floor(n/2)).

A326133 Numbers n for which sigma(n) > A005187(n).

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, 110, 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, 234, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280, 282, 288, 294, 300

Offset: 1

Antti Karttunen, Jun 11 2019

nonn

Differs from A023196 for the first time at the 28th term, which here is 110, which is not included in A023196.

Note that as there is at least one odd number (815634435) in A326138, it means that A005231 does not contain all odd terms of this sequence.

Positions of negative terms in A294898.
Cf. A000203, A005187, A023196.
Cf. A000396, A005231, A083207, A111592, A326131, A326138 (subsequences).

Select[Range[300], DivisorSigma[1, #] > 2*# - DigitCount[2*#, 2, 1] &] (* Amiram Eldar, Aug 06 2023 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326133(n) = (sigma(n)>A005187(n));

A257126 a(n) = A055938(n) - A005187(n).

Original entry on oeis.org

1, 2, 2, 2, 4, 3, 3, 2, 4, 3, 5, 5, 5, 4, 4, 2, 4, 3, 5, 5, 5, 4, 6, 5, 5, 6, 8, 6, 6, 5, 5, 2, 4, 3, 5, 5, 5, 4, 6, 5, 5, 6, 8, 6, 6, 5, 7, 5, 5, 6, 8, 6, 6, 7, 9, 6, 8, 9, 9, 7, 7, 6, 6, 2, 4, 3, 5, 5, 5, 4, 6, 5, 5, 6, 8, 6, 6, 5, 7, 5, 5, 6, 8, 6, 6, 7, 9, 6, 8, 9, 9, 7, 7, 6, 8, 5, 5, 6, 8, 6, 6, 7, 9, 6, 8, 9, 9, 7, 7, 8, 10, 6, 8, 9, 9, 7, 9, 10, 10, 9

Offset: 1

Antti Karttunen, Apr 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Positions of records: A257130.

Cf. A005187, A055938, A256992.

(define (A257126 n) (- (A055938 n) (A005187 n)))

a(n) = A055938(n) - A005187(n).

A265743 a(n) = number of terms of A005187 needed to sum to n using the greedy algorithm.

Original entry on oeis.org

0, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 3, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 1, 1

Offset: 0

Antti Karttunen, Dec 17 2015

nonn

a(0) = 0, because no numbers are needed to form an empty sum, which is zero.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A005187, A055938.

Cf. also A000120, A007895, A053610, A265404, A265744, A265745.

Other identities. For all n >= 1:

a(A005187(n)) = 1 and a(A055938(n)) > 1.

A294896 a(n) = gcd(A000203(n), A005187(n)).

Original entry on oeis.org

1, 3, 4, 7, 2, 2, 1, 15, 1, 18, 1, 2, 1, 1, 2, 31, 2, 1, 5, 2, 1, 1, 6, 2, 1, 7, 10, 1, 6, 8, 1, 63, 16, 6, 1, 7, 1, 1, 2, 6, 1, 3, 2, 1, 2, 8, 1, 2, 19, 1, 2, 1, 6, 8, 3, 1, 10, 2, 1, 4, 1, 1, 8, 127, 4, 2, 1, 2, 3, 1, 6, 1, 1, 1, 2, 1, 6, 8, 1, 2, 1, 7, 6, 1, 2, 12, 1, 1, 6, 2, 1, 12, 1, 3, 8, 2, 1, 1, 2, 1, 6, 8, 1, 5, 2

Offset: 1

Antti Karttunen, Nov 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A005187, A294898, A294899, A295655, A295656.

Array[GCD[DivisorSigma[1, #], IntegerExponent[(2 #)!, 2]] &, 105] (* Michael De Vlieger, Nov 26 2017 *)

(define (A294896 n) (gcd (A000203 n) (A005187 n)))

a(n) = gcd(A000203(n), A005187(n)).
a(n) = gcd(A000203(n),abs(A294898(n))) = gcd(A005187(n),abs(A294898(n))), where A294898(n) = A005187(n) - A000203(n).
a(n) = A000203(n)/A295655(n) = A005187(n)/A295656(n).

A296208 Xor-Moebius transform of A005187.

Original entry on oeis.org

1, 2, 5, 4, 9, 12, 10, 8, 20, 24, 18, 24, 22, 16, 23, 16, 33, 60, 34, 48, 41, 56, 43, 48, 39, 36, 34, 40, 55, 52, 56, 32, 86, 96, 65, 120, 70, 104, 88, 96, 78, 104, 83, 120, 88, 112, 88, 96, 84, 84, 71, 80, 103, 104, 115, 80, 72, 68, 112, 96, 116, 76, 75, 64, 158, 244, 130, 192, 168, 192, 139, 240, 142, 212, 175, 216

Offset: 1

Antti Karttunen, Dec 25 2017

Unique sequence satisfying SumXOR_{d divides n} a(d) = A005187(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. A297111 gives the ordinary Möbius transform of A005187.

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

Cf. A003987, A005187, A256739, A294899, A295901, A296203, A297110, A297111.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A296208(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A005187(d)))); (v); } \\ after code in A295901.

A300723 Möbius-transform of A005187(A053645(n)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 4, 0, 0, 2, 4, 4, 8, 6, 9, 0, 1, 0, 4, 4, 3, 6, 11, 8, 15, 10, 18, 12, 23, 10, 26, 0, -4, 2, -1, 0, 8, 6, 2, 8, 16, 2, 19, 12, 12, 14, 26, 16, 28, 16, 33, 20, 39, 20, 37, 24, 42, 26, 50, 20, 54, 30, 49, 0, -8, -6, 4, 4, -4, -2, 11, 0, 16, 10, -7, 12, 15, 2, 26, 16, 13, 18, 35, 4, 37, 22, 18, 24, 47, 12

Offset: 1

Antti Karttunen, Mar 12 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A008683, A053645, A297111, A300724, A300725.

With[{s = Array[2 # - DigitCount[2 #, 2, 1] &[# - 2^Floor@ Log2@ #] &, 90]}, Table[DivisorSum[n, MoebiusMu[n/#] s[[#]] &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A300723(n) = sumdiv(n,d,moebius(n/d)*A005187(A053645(d)));

a(n) = Sum_{d|n} A008683(n/d)*A005187(A053645(d)).

a(1) = 0; for n > 1, a(n) = A297111(n) - 2*A300724(n).

A317927 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A005187.

Original entry on oeis.org

1, 3, 2, 19, 4, 2, 11, 63, 6, 3, 19, 13, 23, 17, 5, 867, 16, 4, 35, 5, 17, 25, 21, 11, 31, 29, 13, 113, 27, 13, 57, 3069, 13, 9, 23, 25, 71, 41, 14, 69, 79, 33, 41, 169, 9, 25, 89, 615, 259, 53, 17, 197, 51, 25, 29, 389, 20, 31, 113, 59, 117, 67, 10, 22199, 18, 14, 131, 31, 51, 71, 69, 11, 143, 77, 22, 281, 91, 35, 153, 489, 71, 85, 81, 151, 19

Offset: 1

Antti Karttunen, Aug 11 2018

The first negative term is a(330) = -21.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A317928 (denominators).

Cf. also A297111, A300244, A299151, A317931.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A317927perA317928(n) = if(1==n,n,(A005187(n)-sumdiv(n,d,if((d>1)&&(dA317927perA317928(d)*A317927perA317928(n/d),0)))/2);
A317927(n) = numerator(A317927perA317928(n));

\\ Memoized implementation:
memo = Map();
A317927perA317928(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A005187(n)-sumdiv(n,d,if((d>1)&&(dA317927perA317928(d)*A317927perA317928(n/d),0)))/2); mapput(memo,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A005187(n) - Sum_{d|n, d>1, d 1.

A317928 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A005187.

Original entry on oeis.org

1, 2, 1, 8, 1, 1, 2, 16, 1, 1, 2, 4, 2, 4, 1, 128, 1, 1, 2, 1, 2, 4, 1, 2, 2, 4, 1, 16, 1, 2, 2, 256, 1, 1, 2, 4, 2, 4, 1, 8, 2, 4, 1, 16, 1, 2, 2, 64, 8, 4, 1, 16, 1, 2, 2, 32, 1, 2, 2, 8, 2, 4, 1, 1024, 1, 1, 2, 2, 2, 4, 1, 1, 2, 4, 1, 16, 4, 2, 2, 32, 2, 4, 1, 16, 1, 2, 2, 32, 1, 2, 4, 8, 2, 4, 1, 64, 2, 16, 1, 16, 1, 2, 2, 32, 2

Offset: 1

Antti Karttunen, Aug 11 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A317927 (numerators).

Cf. also A297111, A300244, A299152, A317932.

6
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A317927perA317928(n) = if(1==n,n,(A005187(n)-sumdiv(n,d,if((d>1)&&(dA317927perA317928(d)*A317927perA317928(n/d),0)))/2);
A317928(n) = denominator(A317927perA317928(n));

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A005187(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

A279357 a(n) = A005187(n) XOR A005187(n+1).

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A283997 a(n) = n XOR A005187(floor(n/2)), where XOR is bitwise-xor (A003987).

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A326133 Numbers n for which sigma(n) > A005187(n).

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A257126 a(n) = A055938(n) - A005187(n).

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A265743 a(n) = number of terms of A005187 needed to sum to n using the greedy algorithm.

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A294896 a(n) = gcd(A000203(n), A005187(n)).

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A296208 Xor-Moebius transform of A005187.

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A300723 Möbius-transform of A005187(A053645(n)).

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A317927 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A005187.