A323896 -id:A323896 - cp's OEIS frontend

A323365 Sum of Stern's Diatomic sequence, A002487 and its Dirichlet inverse, A317843.

2, 0, 0, 1, 0, 4, 0, 1, 4, 6, 0, 2, 0, 6, 12, 1, 0, 4, 0, 3, 12, 10, 0, 2, 9, 10, 8, 3, 0, -4, 0, 1, 20, 10, 18, 4, 0, 14, 20, 3, 0, 4, 0, 5, 4, 14, 0, 2, 9, 5, 20, 5, 0, 8, 30, 3, 28, 14, 0, 4, 0, 10, 20, 1, 30, -8, 0, 5, 28, 0, 0, 4, 0, 22, -2, 7, 30, 0, 0, 3, 16, 22, 0, 8, 30, 26, 28, 5, 0, 20, 30, 7, 20, 18, 42, 2, 0, 9, 4, 7, 0, 4, 0, 5, 0

Offset: 1

Antti Karttunen, Jan 13 2019

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

Cf. A002487 (also a quadrisection of this sequence), A317843.

Cf. also A317837, A319687, A323885, A323887, A323894, A323896.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317843(n) = if(1==n,1,-sumdiv(n,d,if(dA002487(n/d)*A317843(d),0)));
A323365(n) = (A002487(n) + A317843(n));

a(n) = A002487(n) + A317843(n).
From Antti Karttunen, Dec 08 2021: (Start)
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A002487(d) * A317843(n/d).
a(4*n) = A002487(n).
(End)

A323894 Sum of A048673 and its Dirichlet inverse, A323893.

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 12, 9, 16, 0, 26, 0, 24, 24, 37, 0, 46, 0, 36, 36, 28, 0, 76, 16, 36, 51, 56, 0, 58, 0, 114, 42, 40, 48, 121, 0, 48, 54, 106, 0, 94, 0, 66, 104, 60, 0, 223, 36, 92, 60, 86, 0, 220, 56, 166, 72, 64, 0, 164, 0, 76, 162, 349, 72, 112, 0, 96, 90, 136, 0, 354, 0, 84, 150, 116, 84, 148, 0, 312, 277, 88, 0, 260, 80, 96, 96

Offset: 1

Antti Karttunen, Feb 08 2019

sign

The first four negative terms are a(3063060) = -14126242, a(3423420) = -17546656, a(4084080) = -14460312, a(4144140) = -22677277. - Antti Karttunen, Apr 20 2022

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Cf. A003961, A048673, A323893, A349135.

Cf. also A323365, A323412, A323896, A349135, A349349, A353336.

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(dA003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v323893 = DirInverse(vector(up_to,n,A048673(n)));
A323893(n) = v323893[n];
A323894(n) = (A048673(n)+A323893(n));

a(n) = A048673(n) + A323893(n).
For n > 1, a(n) = -Sum_{d|n, 1A048673(n/d) * A323893(d). - Antti Karttunen, Apr 20 2022
a(n) = A349135(A003961(n)). - Antti Karttunen, Nov 30 2024

A323885 Sum of A001511 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 4, 0, 4, 1, 4, 0, 2, 0, 4, 2, 5, 0, 2, 0, 2, 2, 4, 0, 4, 1, 4, 1, 2, 0, 0, 0, 6, 2, 4, 2, 3, 0, 4, 2, 4, 0, 0, 0, 2, 1, 4, 0, 5, 1, 2, 2, 2, 0, 2, 2, 4, 2, 4, 0, 4, 0, 4, 1, 7, 2, 0, 0, 2, 2, 0, 0, 4, 0, 4, 1, 2, 2, 0, 0, 5, 1, 4, 0, 4, 2, 4, 2, 4, 0, 2, 2, 2, 2, 4, 2, 6, 0, 2, 1, 3, 0, 0, 0, 4, 0

Offset: 1

Antti Karttunen, Feb 08 2019

nonn

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

Cf. A001511, A092673.

Cf. also A318449, A318450, A323365, A323882, A323884, A323887, A323896.

A001511(n) = (1+valuation(n,2));
A092673(n) = (moebius(n)-if(n%2,0,moebius(n/2)));
A323885(n) = (A001511(n)+A092673(n));

from sympy import mobius
def A323885(n): return (n&-n).bit_length()+mobius(n)-(0 if n&1 else mobius(n>>1)) # Chai Wah Wu, Jul 13 2022

a(n) = A001511(n) + A092673(n).

A323887 Sum of Per Nørgård's "infinity sequence" (A004718) and its Dirichlet inverse (A323886).

Original entry on oeis.org

2, 0, 0, 1, 0, -4, 0, -1, 4, 0, 0, 2, 0, -6, 0, 1, 0, 0, 0, 0, 12, -2, 0, -2, 0, 2, 0, 3, 0, -8, 0, -1, 4, 0, 0, 2, 0, -6, -4, 0, 0, 10, 0, 1, 16, -4, 0, 2, 9, -6, 0, -1, 0, 0, 0, -3, 12, 4, 0, 4, 0, -10, -20, 1, 0, 0, 0, 0, 8, -2, 0, -2, 0, 2, 12, 3, 6, -12, 0, 0, -4, -2, 0, 1, 0, -4, -8, -1, 0, 16, -6, 2, 20, -6, 0, -2, 0, 11, 0, 3, 0, -8, 0, 1, 28

Offset: 1

Antti Karttunen, Feb 08 2019

sign

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A004718, A323886.

Cf. also A323365, A323882, A323885, A323896.

up_to = 65537;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA004718list(up_to);
A004718(n) = v004718[n];
v323886 = DirInverse(v004718);
A323886(n) = v323886[n];
A323887(n) = (A004718(n)+A323886(n));

a(n) = A004718(n) + A323886(n).

A323895 Dirichlet inverse of binary Gray code, A003188.

Original entry on oeis.org

1, -3, -2, 3, -7, 7, -4, -3, -9, 27, -14, -10, -11, 15, 20, 3, -25, 35, -26, -45, -15, 55, -28, 13, 28, 43, 22, -24, -19, -91, -16, -3, 7, 99, 6, -58, -55, 103, -8, 63, -61, 55, -62, -94, 71, 111, -56, -16, -25, -148, 58, -73, -47, -107, 152, 33, 67, 75, -38, 196, -35, 63, 148, 3, 57, -53, -98, -171, 9, -59, -100, 78, -109, 219, -208, -178, 5, 13

Offset: 1

Antti Karttunen, Feb 08 2019

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

Cf. A003188, A323896.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003188(n) = bitxor(n, n>>1);
v323895 = DirInverseCorrect(vector(up_to,n,A003188(n)));
A323895(n) = v323895[n];

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A323365 Sum of Stern's Diatomic sequence, A002487 and its Dirichlet inverse, A317843.

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A323894 Sum of A048673 and its Dirichlet inverse, A323893.

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A323885 Sum of A001511 and its Dirichlet inverse.

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A323887 Sum of Per Nørgård's "infinity sequence" (A004718) and its Dirichlet inverse (A323886).

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A323895 Dirichlet inverse of binary Gray code, A003188.

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