A297111 -id:A297111 - cp's OEIS frontend

A346237 Dirichlet inverse of A005187.

1, -3, -4, 2, -8, 14, -11, 0, 0, 30, -19, -14, -23, 41, 38, 0, -32, -2, -35, -34, 49, 73, -42, 4, 17, 89, 14, -46, -54, -172, -57, 0, 88, 126, 109, 10, -71, 137, 110, 12, -79, -219, -82, -86, -6, 164, -89, 0, 26, -103, 158, -106, -102, -76, 199, 16, 170, 212, -113, 274, -117, 223, 16, 0, 240, -406, -131, -154, 201

Offset: 1

Antti Karttunen, Jul 13 2021

sign

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

Cf. A005187, A346238.

Cf. also A297111, A317927, A317928.

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];

a(n) = A346238(n) - A005187(n).

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).

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

A378991 Dirichlet inverse of the Möbius transform of A005187, where A005187(n) = 2*n - (number of 1's in binary representation of n).

Original entry on oeis.org

1, -2, -3, 0, -7, 8, -10, 0, -3, 20, -18, -4, -22, 28, 27, 0, -31, 6, -34, -12, 35, 52, -41, 0, 10, 64, 11, -16, -53, -104, -56, 0, 66, 92, 91, 4, -70, 100, 84, 0, -78, -132, -81, -32, 21, 120, -88, 0, 16, -66, 123, -40, -101, -56, 173, 0, 132, 156, -112, 124, -116, 164, 51, 0, 210, -256, -130, -60, 156, -364, -137

Offset: 1

Antti Karttunen, Dec 15 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..20020

Dirichlet inverse of A297111.
Inverse Möbius transform of A346237.
Cf. A005187.
Cf. also A378989, A378990.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));
memoA378991 = Map();
A378991(n) = if(1==n,1,my(v); if(mapisdefined(memoA378991,n,&v), v, v = -sumdiv(n,d,if(dA297111(n/d)*A378991(d),0)); mapput(memoA378991,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA297111(n/d) * a(d).

a(n) = Sum_{d|n} A346237(d).

cp's OEIS Frontend

A346237 Dirichlet inverse of A005187.

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

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Formula

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.

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A378991 Dirichlet inverse of the Möbius transform of A005187, where A005187(n) = 2*n - (number of 1's in binary representation of n).

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Author

Keywords

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PARI

Formula

cp's OEIS Frontend

A346237 Dirichlet inverse of A005187.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A378991 Dirichlet inverse of the Möbius transform of A005187, where A005187(n) = 2*n - (number of 1's in binary representation of n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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.