A323885 -id:A323885 - 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)

A323882 Sum of A126760 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, 1, 1, 4, 0, 1, 0, 6, 4, 1, 0, 1, 0, 2, 6, 8, 0, 1, 4, 10, 1, 3, 0, 0, 0, 1, 8, 12, 12, 1, 0, 14, 10, 2, 0, 0, 0, 4, 2, 16, 0, 1, 9, 14, 12, 5, 0, 1, 16, 3, 14, 20, 0, 2, 0, 22, 3, 1, 20, 0, 0, 6, 16, 12, 0, 1, 0, 26, 14, 7, 24, 0, 0, 2, 1, 28, 0, 3, 24, 30, 20, 4, 0, 2, 30, 8, 22, 32, 28, 1, 0, 25, 4, 9, 0, 0, 0, 5, 12

Offset: 1

Antti Karttunen, Feb 08 2019

From Antti Karttunen, Aug 18 2021: (Start)
No negative terms in range 1 .. 2^20.
Apparently zeros occur only on (some of the) positions given by A030059, with exceptions for example on n = 70, 105, 110, 130, 154, etc, where a(n) > 0.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A030059, A126760, A323881, A323884, A323885, A323887.

up_to = 20000;
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1])*sumdiv(n, d, if(dA126760(n)));
A323881(n) = v323881[n];
A323882(n) = (A126760(n)+A323881(n));

a(n) = A126760(n) + A323881(n).

For n > 1, a(n) = -Sum_{d|n, 1A126760(d) * A323881(n/d). - Antti Karttunen, Aug 18 2021

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).

A323896 Sum of binary Gray code A003188 and its Dirichlet inverse, A323895.

Original entry on oeis.org

2, 0, 0, 9, 0, 12, 0, 9, 4, 42, 0, 0, 0, 24, 28, 27, 0, 62, 0, -15, 16, 84, 0, 33, 49, 66, 44, -6, 0, -74, 0, 45, 56, 150, 56, -4, 0, 156, 44, 123, 0, 118, 0, -36, 130, 168, 0, 24, 16, -105, 100, -27, 0, -62, 196, 69, 104, 114, 0, 230, 0, 96, 180, 99, 154, 46, 0, -69, 112, 42, 0, 186, 0, 330, -98, -72, 112, 118, 0, 39, 117, 366, 0, 47

Offset: 1

Antti Karttunen, Feb 08 2019

sign

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

Cf. A003188, A323895.

Cf. also A323365, A323412, A323885, A323887, A323894.

up_to = 65537;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA003188(n) = bitxor(n, n>>1);
v323895 = DirInverse(vector(up_to,n,A003188(n)));
A323895(n) = v323895[n];
A323896(n) = (A003188(n)+A323895(n));

a(n) = A003188(n) + A323895(n).

A323884 Sum of A322026 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 8, 9, 4, 0, 8, 0, 4, 6, 8, 0, 4, 0, 4, 6, 4, 0, 0, 1, 4, 15, 4, 0, -2, 0, 12, 6, 4, 2, 13, 0, 4, 6, 4, 0, -2, 0, 4, 5, 4, 0, 22, 1, 2, 6, 4, 0, 7, 2, 4, 6, 4, 0, 8, 0, 4, 5, 20, 2, -2, 0, 4, 6, 0, 0, 38, 0, 4, 3, 4, 2, -2, 0, 10, 13, 4, 0, 8, 2, 4, 6, 4, 0, 16, 2, 4, 6, 4, 2, 28, 0, 2, 5, 4, 0, -2, 0, 4, 0

Offset: 1

Antti Karttunen, Feb 08 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007814, A007949, A322026, A323882, A323883, A323885.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
A322026(n) = v322026[n];
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA323883(n) = v323883[n];
A323884(n) = (A322026(n)+A323883(n));

a(n) = A322026(n) + A323883(n).

A323900 Sum of A287896 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 8, 0, 4, 4, 12, 0, 4, 0, 12, 12, 5, 0, 8, 0, 6, 12, 20, 0, 8, 9, 20, 8, 6, 0, -8, 0, 6, 20, 20, 18, 12, 0, 28, 20, 12, 0, 8, 0, 10, 4, 28, 0, 10, 9, 10, 20, 10, 0, 16, 30, 12, 28, 28, 0, 20, 0, 20, 20, 7, 30, -16, 0, 10, 28, 0, 0, 16, 0, 44, -2, 14, 30, 0, 0, 15, 16, 44, 0, 28, 30, 52, 28, 20, 0, 40

Offset: 1

Antti Karttunen, Feb 12 2019

sign

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

Cf. A001511, A002487, A287896, A323365, A323885, A323899.

up_to = 20000;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001511(n) = (1+valuation(n,2));
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A287896(n) = (A001511(n)*A002487(n));
v323899 = DirInverse(vector(up_to,n,A287896(n)));
A323899(n) = v323899[n];
A323900(n) = (A287896(n)+A323899(n));

a(n) = A287896(n) + A323899(n).

A346488 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(n) = 0 if mu(n) = -1, and f(n) = n for all other numbers (with mu = Möbius mu, A008683).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 2, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 2, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 38, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 2, 2, 47, 48, 2, 2, 49, 2, 50, 51, 52, 53, 2, 2, 54, 55, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71

Offset: 1

Antti Karttunen, Aug 20 2021

nonn

Restricted growth sequence transform of the sequence f(n) = 0 if mu(n) = -1, and f(n) = n for mu(n) >= 0.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j) => A305980(i) = A305980(j),
  a(i) = a(j) => b(i) = b(j), where b is the pointwise sum of any two multiplicative sequences c and d that are Dirichlet inverses of each other. For example, b can be a sequence like A319340, A323885, or A347094.

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

Cf. A008683, A070549, A030059 (positions of 2's).

Cf. also A305800, A305980, A319340, A323885, A347094.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux346488(n) = if(moebius(n)<0,0,n);
v346488 = rgs_transform(vector(up_to, n, Aux346488(n)));
A346488(n) = v346488[n];

A070549(n) = sum(k=1,n,(-1==moebius(k)));
A346488(n) = if(1==n,1,if(-1==moebius(n),2,1+n-A070549(n)));

a(1) = 1, and for n > 1, if A008683(n) = -1, a(n) = 2, otherwise a(n) = 1 + n - A070549(n).

cp's OEIS Frontend

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

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A323882 Sum of A126760 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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A323896 Sum of binary Gray code A003188 and its Dirichlet inverse, A323895.

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A323884 Sum of A322026 and its Dirichlet inverse.

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A323900 Sum of A287896 and its Dirichlet inverse.

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A346488 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, where f(n) = 0 if mu(n) = -1, and f(n) = n for all other numbers (with mu = Möbius mu, A008683).

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