A300244 -id:A300244 - cp's OEIS frontend

A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

1, 2, 3, 4, 7, 4, 10, 8, 12, 8, 18, 8, 22, 12, 15, 16, 31, 12, 34, 16, 25, 20, 41, 16, 39, 24, 34, 24, 53, 16, 56, 32, 42, 32, 49, 24, 70, 36, 48, 32, 78, 24, 81, 40, 48, 44, 88, 32, 84, 40, 63, 48, 101, 36, 79, 48, 72, 56, 112, 32, 116, 60, 69, 64, 98, 40, 130, 64, 90, 48, 137, 48, 142, 72, 81, 72, 121, 48, 152, 64

Offset: 1

Antti Karttunen, Dec 25 2017

nonn

Sequence differs from A035532 for the first time at n = 15, 21, 25, 27, 33, 35, 51, etc., i.e., at those composite n where A297115 has a nonzero value. - Antti Karttunen & M. F. Hasler, Mar 10 2018

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

Cf. A000010, A005187, A008683, A297108, A297110, A297114, A297115, A297117, A300244, A300723, A300724, A300725.

Table[DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 80}] (* Michael De Vlieger, Mar 10 2018 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));

a(n) = Sum_{d|n} A005187(d)*A008683(n/d).
a(n) = n + A297114(n).
From Antti Karttunen, Mar 11 2018: (Start)
Sum A005187(n) x^n = Sum a(n)*x^n/(1-x^n). [Another way of saying that this is the Möbius transform of A005187. This was originally included in A035532 by mistake.]
a(n) = 2*phi(n) - A297115(n) = phi(n) + A297117(n).
a(n) = A005187(n) - A300244(n).
a(1) = 1; for n > 1, a(n) = A300723(n) + 2*A300724(n).
(End)

A317844 Difference between A294898 and its Möbius transform (A297114).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 3, 2, 0, 0, 1, 0, 0, 3, 7, 0, -6, 2, 9, 3, 1, 0, -2, 0, 0, 7, 14, 5, -9, 0, 15, 9, -4, 0, 3, 0, 5, 5, 18, 0, -14, 3, 14, 14, 7, 0, 2, 9, -3, 15, 24, 0, -24, 0, 25, 10, 0, 11, 12, 0, 12, 18, 15, 0, -29, 0, 33, 16, 13, 10, 14, 0, -12, 10, 37, 0, -23, 16, 38, 24, 1, 0, -16, 12, 16, 25

Offset: 1

Antti Karttunen, Aug 09 2018

sign

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

Cf. A000203, A001065, A005187, A008683, A294898, A297114, A300244.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A317844(n) = -sumdiv(n,d,(dA005187(d)-sigma(d)));

a(n) = -Sum_{d|n, dA008683(n/d)*A294898(d).
a(n) = A294898(n) - A297114(n).
a(n) = A300244(n) - A001065(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.

A318446 Inverse Möbius transform of A005187: a(n) = Sum_{d|n} A005187(d).

Original entry on oeis.org

1, 4, 5, 11, 9, 18, 12, 26, 21, 30, 20, 47, 24, 40, 39, 57, 33, 68, 36, 75, 55, 64, 43, 108, 56, 76, 71, 100, 55, 126, 58, 120, 88, 102, 87, 167, 72, 112, 102, 168, 80, 174, 83, 156, 141, 134, 90, 233, 107, 174, 135, 184, 103, 222, 133, 224, 150, 170, 114, 309, 118, 180, 191, 247, 160, 272, 132, 243, 182, 270, 139, 370, 144

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

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

Cf. A005187, A318445.

Cf. also A297111, A300244.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A318446(n) = sumdiv(n,d,A005187(d));

a(n) = Sum_{d|n} A005187(d).
a(n) = A005187(n) + A318445(n).
a(n) = A318448(n) + A007429(n).

A300726 Difference between A053644 (the largest power of 2 less than or equal to n) and its Möbius transform.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 5, 5, 8, 1, 10, 1, 10, 5, 9, 1, 12, 4, 9, 8, 10, 1, 13, 1, 16, 9, 17, 7, 20, 1, 17, 9, 20, 1, 21, 1, 18, 14, 17, 1, 24, 4, 20, 17, 18, 1, 24, 11, 20, 17, 17, 1, 26, 1, 17, 22, 32, 11, 41, 1, 34, 17, 39, 1, 40, 1, 33, 20, 34, 11, 41, 1, 40, 16, 33, 1, 42, 19, 33, 17, 36, 1, 46, 11, 34, 17, 33, 19

Offset: 1

Antti Karttunen, Mar 11 2018

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

Cf. A008683, A053644, A300724, A300725, A300244.

With[{s = Array[2^Floor@ Log2@ # &, 95]}, Table[s[[n]] - 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
A300726(n) = -sumdiv(n,d,(dA053644(d));

a(n) = -Sum_{d|n, dA008683(n/d)*A053644(d).

a(n) = A053644(n) - A300724(n).

A346238 Sum of A005187 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 15, 16, 48, 0, 8, 0, 66, 64, 31, 0, 32, 0, 4, 88, 114, 0, 50, 64, 138, 64, 7, 0, -116, 0, 63, 152, 192, 176, 80, 0, 210, 184, 90, 0, -138, 0, -1, 80, 252, 0, 94, 121, -6, 256, -5, 0, 28, 304, 125, 280, 324, 0, 390, 0, 342, 136, 127, 368, -276, 0, -20, 336, -386, 0, 126, 0, 426, 24, -17, 418, -360, 0

Offset: 1

Antti Karttunen, Jul 13 2021

sign

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

Cf. A005187, A346237.

Cf. also A300244.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
v346237 = DirInverseCorrect(vector(up_to,n,A005187(n)));
A346237(n) = v346237[n];
A346238(n) = (A005187(n)+A346237(n));

a(n) = A005187(n) + A346237(n).

cp's OEIS Frontend

A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

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A317844 Difference between A294898 and its Möbius transform (A297114).

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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.

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A318446 Inverse Möbius transform of A005187: a(n) = Sum_{d|n} A005187(d).

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A300726 Difference between A053644 (the largest power of 2 less than or equal to n) and its Möbius transform.

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A346238 Sum of A005187 and its Dirichlet inverse.

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