A317927 -id:A317927 - cp's OEIS frontend

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

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.

A346237 Dirichlet inverse of A005187.

Original entry on oeis.org

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).

A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 3, 5, 3, 3, 5, 3, 5, 3, 1, 35, 5, 3, 7, 9, 5, 5, 7, 5, 19, 5, 5, 9, 7, 1, 5, 63, 1, 5, 9, 9, 11, 7, 5, 15, 11, 5, 13, 15, 13, 7, 9, 35, 27, 19, 7, 15, 13, 5, 7, 15, 3, 7, 11, 3, 9, 5, -7, 231, -1, 1, 11, 15, 7, 9, 13, 15, 15, 11, 47, 21, 19, 5, 13, 105, 27, 11, 19, 15, 27, 13, 11, 25, 17, 13, 23, 21, 11, 9, 1, 63

Offset: 1

Antti Karttunen, Aug 11 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Stern's sequences

Cf. A002487, A317932 (denominators, conjectured).

Cf. also A299151, A317927, A317839, A317843.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317931perA317932(n) = if(1==n,n,(A002487(n)-sumdiv(n,d,if((d>1)&&(dA317931perA317932(d)*A317931perA317932(n/d),0)))/2);
A317931(n) = numerator(A317931perA317932(n));

\\ Memoized implementation:
memo = Map();
A317931perA317932(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A002487(n)-sumdiv(n,d,if((d>1)&&(dA317931perA317932(d)*A317931perA317932(n/d),0)))/2); mapput(memo,n,v); (v)));

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A002487(n) - Sum_{d|n, d>1, d 1.

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A346237 Dirichlet inverse of A005187.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula