A323365 -id:A323365 - cp's OEIS frontend

A323894 Sum of A048673 and its Dirichlet inverse, A323893.

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).

A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 9, 16, 36, 0, 12, 0, 48, 48, 27, 0, 24, 0, 18, 64, 72, 0, 60, 36, 84, 32, 24, 0, 0, 0, 45, 96, 108, 96, 84, 0, 120, 112, 90, 0, 0, 0, 36, 48, 144, 0, 84, 64, 72, 144, 42, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 64, 99, 168, 0, 0, 54, 192, 0, 0, 132, 0, 228, 96, 60, 192, 0, 0, 126, 112, 252, 0, 288, 216, 264, 240, 180, 0

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

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

Cf. A001615, A323363.

Cf. also A319340, A322581, A323365, A323372, A323399.

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));
A323364(n) = (A001615(n) + A323363(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).

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).

A323412 Sum of the inverse permutation of EKG-sequence, A064664, and its Dirichlet inverse, A323411.

Original entry on oeis.org

2, 0, 0, 4, 0, 20, 0, 4, 25, 40, 0, -14, 0, 56, 100, 21, 0, -86, 0, -24, 140, 80, 0, 64, 100, 112, -65, -32, 0, -386, 0, 30, 200, 132, 280, 233, 0, 148, 280, 138, 0, -538, 0, -44, -520, 172, 0, -55, 196, -324, 330, -60, 0, 596, 400, 194, 370, 228, 0, 898, 0, 244, -732, 67, 560, -766, 0, -70, 430, -1068, 0, -380, 0, 268, -1040, -78, 560

Offset: 1

Antti Karttunen, Jan 13 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to EKG sequence

Cf. A064664, A323411.

Cf. also A304527, A323365.

A323412(n) = (A064664(n)+A323411(n)); \\ Uses also the program code given in A323411.

A349349 Sum of A252463 and its Dirichlet inverse, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 3, 4, 6, 0, 8, 0, 10, 12, 7, 0, 8, 0, 13, 20, 14, 0, 15, 9, 22, 8, 19, 0, 14, 0, 15, 28, 26, 30, 19, 0, 34, 44, 25, 0, 18, 0, 29, 12, 38, 0, 28, 25, 21, 52, 37, 0, 24, 42, 35, 68, 46, 0, 28, 0, 58, 20, 31, 66, 30, 0, 47, 76, 32, 0, 38, 0, 62, 18, 55, 70, 30, 0, 47, 16, 74, 0, 36, 78, 82, 92, 55

Offset: 1

Antti Karttunen, Nov 15 2021

nonn

Question: Are there any negative terms? All terms in range 1 .. 2^23 are nonnegative. (See also A349126). - Antti Karttunen, Apr 20 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Coincides with A349126 on odd numbers.
Cf. A064989, A252463, A349348.
Cf. also A323365, A323412, A323894, A349135, A353336.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
v349348 = DirInverseCorrect(vector(up_to,n,A252463(n)));
A349348(n) = v349348[n];
A349349(n) = (A252463(n)+A349348(n));

a(n) = A252463(n) + A349348(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A252463(d) * A349348(n/d).
For all n >= 1, a(2n-1) = A349126(2n-1).

A319687 a(n) = A318509(n) - A002487(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 4, 0, 4, 0, 0, 2, 0, 0, 6, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, -2, -6, 0, -4, 0, 0, 0, -4, 0, -6, 0, 10, 0, 0, 0, 4, 2, 0, -2, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 02 2018

All terms seem to be even. See the conjecture given in A261179.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A002487, A261179, A317837, A318509, A323365.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318509(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = A002487(f[i, 1])); factorback(f); };
A319687(n) = (A318509(n) - A002487(n));

from math import prod
from functools import reduce
from sympy import factorint
def A319687(n): return prod(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(p)[-1:2:-1],(1,0)))**e for p, e in factorint(n).items())-sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) # Chai Wah Wu, May 18 2023

a(n) = A318509(n) - A002487(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).

cp's OEIS Frontend

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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A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

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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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A323412 Sum of the inverse permutation of EKG-sequence, A064664, and its Dirichlet inverse, A323411.

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A349349 Sum of A252463 and its Dirichlet inverse, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

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A319687 a(n) = A318509(n) - A002487(n).

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A323900 Sum of A287896 and its Dirichlet inverse.

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