A297111 -id:A297111 - cp's OEIS frontend

A300244 Difference between A005187 and its Möbius transform (A297111).

0, 1, 1, 3, 1, 6, 1, 7, 4, 10, 1, 14, 1, 13, 11, 15, 1, 22, 1, 22, 14, 21, 1, 30, 8, 25, 16, 29, 1, 40, 1, 31, 22, 34, 18, 46, 1, 37, 26, 46, 1, 57, 1, 45, 38, 44, 1, 62, 11, 57, 35, 53, 1, 68, 26, 61, 38, 56, 1, 84, 1, 59, 51, 63, 30, 90, 1, 70, 45, 89, 1, 94, 1, 73, 65, 77, 29, 104, 1, 94, 50, 81, 1, 117, 39, 84, 57, 93, 1, 128, 33, 92, 60, 91, 42

Offset: 1

Antti Karttunen, Mar 10 2018

nonn

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

Cf. A005187, A008683, A297111, A297114, A297117.

Table[IntegerExponent[(2 n)!, 2] - DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 95}] (* or *)
Fold[Function[{a, n}, Append[a, {Abs@ Total@ Map[MoebiusMu[n/#] a[[#, -1]] &, Most@ Divisors@ n], IntegerExponent[(2 n)!, 2]}]], {{0, 1}}, Range[2, 95]][[All, 1]] (* Michael De Vlieger, Mar 10 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A300244(n) = -sumdiv(n,d,(dA005187(d));

a(n) = A005187(n) - A297111(n).

a(n) = -Sum_{d|n, dA008683(n/d)*A005187(d).

A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, -2, 7, -4, 9, -2, 0, 0, 14, -6, 15, -4, 4, -2, 18, -8, 14, -2, 7, -4, 24, -14, 25, 0, 9, -2, 14, -12, 33, -2, 9, -8, 37, -18, 38, -4, 3, -2, 41, -16, 35, -10, 12, -4, 48, -18, 24, -8, 15, -2, 53, -28, 55, -2, 6, 0, 33, -26, 63, -4, 21, -22, 66, -24, 69, -2, 6, -4, 44, -30, 73, -16, 28, -2

Offset: 1

Antti Karttunen, Dec 26 2017

sign

Cf. A000203, A005187, A051953, A083254, A294898, A297111, A297115, A297117, A317844, A324397.

Cf. A378986 (even bisection, -2*A176095).

Table[DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &], {n, 82}] (* Michael De Vlieger, Mar 11 2019 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));

a(n) = Sum_{d|n} A008683(n/d)*A294898(d).
a(n) = A297111(n) - n.
a(n) = A297117(n) - A051953(n).
a(n) = A083254(n) - A297115(n).
a(2n) = A083254(2n) = A378986(n) = -2*A176095(n).
a(n) = A294898(n) - A317844(n).

A297117 Möbius transform of A011371, n minus (number of 1's in binary expansion of n).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 4, 4, 6, 4, 8, 4, 10, 6, 7, 8, 15, 6, 16, 8, 13, 10, 19, 8, 19, 12, 16, 12, 25, 8, 26, 16, 22, 16, 25, 12, 34, 18, 24, 16, 38, 12, 39, 20, 24, 22, 42, 16, 42, 20, 31, 24, 49, 18, 39, 24, 36, 28, 54, 16, 56, 30, 33, 32, 50, 20, 64, 32, 46, 24, 67, 24, 70, 36, 41, 36, 61, 24, 74, 32, 55, 40, 79, 24

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

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

Cf. A000010, A000120, A008683, A011371, A051953, A297111, A297114, A297115.

A297117(n) = sumdiv(n,d,moebius(n/d)*(d-hammingweight(d)));

a(n) = Sum_{d|n} A008683(n/d) * (d - A000120(d)).
a(n) = A297111(n) - A000010(n).
a(n) = A297114(n) + A051953(n).

A300724 Möbius transform of A053644(n), largest power of 2 less than or equal to n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 4, 6, 3, 7, 2, 7, 3, 3, 8, 15, 6, 15, 6, 11, 7, 15, 4, 12, 7, 8, 6, 15, 3, 15, 16, 23, 15, 25, 12, 31, 15, 23, 12, 31, 11, 31, 14, 18, 15, 31, 8, 28, 12, 15, 14, 31, 8, 21, 12, 15, 15, 31, 6, 31, 15, 10, 32, 53, 23, 63, 30, 47, 25, 63, 24, 63, 31, 44, 30, 53, 23, 63, 24, 48, 31, 63, 22, 45, 31, 47, 28, 63, 18, 53, 30, 47, 31

Offset: 1

Antti Karttunen, Mar 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A053644, A297111, A297115, A300723, A300725, A300726.

A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A300724(n) = sumdiv(n,d,moebius(n/d)*A053644(d));

a(n) = Sum_{d|n} A008683(n/d)*A053644(d).

a(n) + A300725(n) = A000010(n).

A300725 Möbius transform of A053645(n), distance to the largest power of 2 less than or equal to n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 0, 0, 1, 3, 2, 5, 3, 5, 0, 1, 0, 3, 2, 1, 3, 7, 4, 8, 5, 10, 6, 13, 5, 15, 0, -3, 1, -1, 0, 5, 3, 1, 4, 9, 1, 11, 6, 6, 7, 15, 8, 14, 8, 17, 10, 21, 10, 19, 12, 21, 13, 27, 10, 29, 15, 26, 0, -5, -3, 3, 2, -3, -1, 7, 0, 9, 5, -4, 6, 7, 1, 15, 8, 6, 9, 19, 2, 19, 11, 9, 12, 25, 6, 19, 14, 13, 15, 27

Offset: 1

Antti Karttunen, Mar 11 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A008683, A053644, A053645, A297111, A297115, A300723, A300724.

With[{s = Array[# - 2^Floor@ Log2@ # &, 95]}, Table[DivisorSum[n, MoebiusMu[n/#] s[[#]] &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A300725(n) = sumdiv(n,d,moebius(n/d)*A053645(d));

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

a(n) + A300724(n) = A000010(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.

A324397 a(1) = 0; for n > 1, a(n) = A297114(A156552(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 3, -2, 3, 0, 7, 0, 14, -2, 0, 0, 9, 0, 15, -6, 9, 0, 18, -4, 33, -2, 14, 0, 4, 0, 25, -2, 42, -4, 7, 0, 254, -26, 9, 0, 33, 0, 63, -2, 140, 0, 41, -8, 14, -34, 127, 0, 24, -12, 66, -90, 579, 0, 38, 0, 684, -2, 6, -4, 21, 0, 175, -2, 37, 0, 24, 0, 3587, -2, 304, -8, 85, 0, 73, -14, 2733, 0, 6, -52, 8707, -378, 11, 0, 3

Offset: 1

Antti Karttunen, Mar 05 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..4137

Cf. A000203, A005187, A156552, A297114, A323243, A323244, A324103, A323247, A323248, A324396, A324398, A324542.

Array[If[# == 1, 0, Function[n, DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &]]@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]]] &, 90] (* Michael De Vlieger, Mar 11 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))));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
\\ Slow: A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));
A297114(n) = (A297111(n) - n);
A324397(n) = if(1==n,0,A297114(A156552(n)));

a(1) = 0; for n > 1, a(n) = A297114(A156552(n)).

For all n >= 1, a(2n-1) = A324103(2n-1).

cp's OEIS Frontend

A300244 Difference between A005187 and its Möbius transform (A297111).

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A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

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A297117 Möbius transform of A011371, n minus (number of 1's in binary expansion of n).

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A300724 Möbius transform of A053644(n), largest power of 2 less than or equal to n.

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A300725 Möbius transform of A053645(n), distance to the largest power of 2 less than or equal to 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.

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