A297115 -id:A297115 - cp's OEIS frontend

A297116 Odd bisection of A297115, Möbius transform of A000120 (binary weight of n).

1, 1, 1, 2, 0, 2, 2, 1, 1, 2, -1, 3, 1, 2, 3, 4, -2, -1, 2, 0, 2, 3, 0, 4, 0, 1, 3, 1, 0, 4, 4, 3, -2, 2, -2, 3, 2, -1, -1, 4, -1, 3, 1, 0, 3, 0, -1, 2, 2, 2, 3, 4, 0, 4, 4, 2, 3, 0, 1, 2, 2, 2, 3, 6, -3, 2, -2, -2, 2, 3, -2, 0, -2, 1, 3, 4, 0, -1, 4, 1, -3, 3, -1, 4, 1, 1, 4, 2, -2, 4, 4, 0, 1, 2, -2, 6, 2, 0, 3, 4, 0, -1, 1, 3, -1

Offset: 1

Antti Karttunen, Dec 26 2017

sign

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

Cf. A000120, A297115.

A297115(n) = sumdiv(n,d,moebius(n/d)*hammingweight(d));
A297116(n) = A297115(n+n-1);

a(n) = A297115(2n - 1).

A083254 a(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 0, 1, 0, 3, -2, 5, 0, 3, -2, 9, -4, 11, -2, 1, 0, 15, -6, 17, -4, 3, -2, 21, -8, 15, -2, 9, -4, 27, -14, 29, 0, 7, -2, 13, -12, 35, -2, 9, -8, 39, -18, 41, -4, 3, -2, 45, -16, 35, -10, 13, -4, 51, -18, 25, -8, 15, -2, 57, -28, 59, -2, 9, 0, 31, -26, 65, -4, 19, -22, 69, -24, 71, -2, 5, -4, 43, -30, 77, -16, 27, -2, 81, -36, 43, -2, 25

Offset: 1

Labos Elemer, May 08 2003

Möbius transform of A033879, deficiency of n. - Antti Karttunen, Dec 26 2017

			Case 1# - totient(x)-cototient[x] = 0 if x is a power of 2;
Case 2# - totient(x)>cototient[x] gives odd primes and also A067800, (= A014076 except probably A036798); e.g. n = 33: a(33) = 2.20-33 = 7; n = p prime: a(p) = p-2;
Case 3# - totient(x)<cototient[x] gives even numbers without powers of 2 and most probably A036798; e.g. n = 20: a(20) = -4; n = 105: a(105) = 2.48-105 = 96-105 = -9.

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A000010, A051953, A000079, A014076, A033879, A067800, A036798, A083255, A115405, A293516, A297114, A297115, A297153, A297154.

A083254 := proc(n)
    2*numtheory[phi](n)-n ;
end proc: # R. J. Mathar, Jan 13 2014

Mathematica
```
Table[2*EulerPhi[w]-w, {w, 1, 1000}]
```

a(n)=2*eulerphi(n)-n \\ Charles R Greathouse IV, Feb 21 2013

a(n) = totient(n) - cototient(n) = A000010(n) - A051953(n).
From Antti Karttunen, Dec 26 2017: (Start)
a(n) = A065620(A297153(n)) = A117966(A297154(n)).
a(n) = A297114(n) + A297115(n).
a(2n) = A297114(2n).
For all n >= 1, -a(A000010(n)) = A293516(n).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where  c = 6/Pi^2 - 1/2 = 0.107927... . - Amiram Eldar, Sep 07 2023

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

Original entry on oeis.org

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)

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).

A378989 Dirichlet inverse of the Möbius transform of binary weight of n.

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -2, 0, 1, 0, -2, 0, -2, 0, 1, 0, -1, 0, -2, 0, 5, 0, -3, 0, 0, 0, -3, 0, -3, 0, -4, 0, 6, 0, 5, 0, -2, 0, 4, 0, -2, 0, -3, 0, -1, 0, -4, 0, 4, 0, 1, 0, -3, 0, 3, 0, 4, 0, -4, 0, -4, 0, -11, 0, 6, 0, -2, 0, 8, 0, -3, 0, -2, 0, 2, 0, 9, 0, -4, 0, 6, 0, -3, 0, 1, 0, 6, 0, -3, 0, 8, 0, 9, 0, 2, 0, -2, 0, -12, 0, -3, 0, -4, 0, -12

Offset: 1

Antti Karttunen, Dec 15 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Dirichlet inverse of A297115.
Inverse Möbius transform of A378990.
Cf. A000120.

A297115(n) = sumdiv(n, d, moebius(n/d)*hammingweight(d));
memoA378989 = Map();
A378989(n) = if(1==n,1,my(v); if(mapisdefined(memoA378989,n,&v), v, v = -sumdiv(n,d,if(dA297115(n/d)*A378989(d),0)); mapput(memoA378989,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA297115(n/d) * a(d).

a(n) = Sum_{d|n} A378990(d).

A035532 a(n) = 2phi(n) if n composite, or 2phi(n) - (A000120(n)-1) if n prime, where phi = A000010, Euler's totient function, and a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 7, 4, 10, 8, 12, 8, 18, 8, 22, 12, 16, 16, 31, 12, 34, 16, 24, 20, 41, 16, 40, 24, 36, 24, 53, 16, 56, 32, 40, 32, 48, 24, 70, 36, 48, 32, 78, 24, 81, 40, 48, 44, 88, 32, 84, 40, 64, 48, 101, 36, 80, 48, 72, 56, 112, 32, 116, 60, 72, 64, 96, 40, 130, 64, 88, 48, 137

Offset: 1

Daniele Parisse

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A010051, A035531, A048881, A297111, A297115.

a035532 1 = 1
a035532 n = if a010051' n == 0 then phi2 else phi2 - a000120 n + 1
            where phi2 = 2 * a000010 n
-- Reinhard Zumkeller, Feb 04 2015

Insert[Table[If[PrimeQ[n],2*EulerPhi[n] - DigitCount[n, 2][[1]] + 1, 2*EulerPhi[n]], {n, 2, 100}], 1, 1] (* Stefan Steinerberger, Apr 11 2006 *)

A035532(n)=2*eulerphi(n)-if(isprime(n),hammingweight(n)-1,n==1) \\ M. F. Hasler, Mar 10 2018

a(n) = 2*A000010(n) - A010051(n)*A048881(n-1), for n > 1. - Reinhard Zumkeller, Feb 04 2015, edited by M. F. Hasler, Mar 10 2018

For many values of n, the inverse Möbius transform of this sequence (g.f.: Sum a(n)*x^n/(1-x^n)) equals A005187, but this is not the case for composite n such that A297115(n) <> 0. The equality does hold for A297111 instead. - Antti Karttunen & M. F. Hasler, Mar 10 2018

More terms from James Sellers

Definition amended for a(1) = 1 by M. F. Hasler, Mar 10 2018

cp's OEIS Frontend

A297116 Odd bisection of A297115, Möbius transform of A000120 (binary weight of n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A083254 a(n) = 2*phi(n) - n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A378989 Dirichlet inverse of the Möbius transform of binary weight of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A035532 a(n) = 2phi(n) if n composite, or 2phi(n) - (A000120(n)-1) if n prime, where phi = A000010, Euler's totient function, and a(1) = 1.