A353336 -id:A353336 - 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

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).

A353335 Dirichlet inverse of A353420.

Original entry on oeis.org

1, -1, -2, 0, -3, 2, -4, 0, -5, 3, -5, 0, -6, 4, 0, 0, -7, 5, -8, 0, -3, 5, -10, 0, -8, 6, -14, 0, -11, 0, -13, 0, -2, 7, -2, 0, -14, 8, -5, 0, -15, 3, -16, 0, 7, 10, -18, 0, -25, 8, -4, 0, -20, 14, -1, 0, -7, 11, -21, 0, -23, 13, 8, 0, -4, 2, -24, 0, -9, 2, -25, 0, -27, 14, 4, 0, -8, 5, -28, 0, -52, 15, -30, 0, -3

Offset: 1

Antti Karttunen, Apr 20 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A126760, A353420, 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); };
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A353420(n) = A126760(A003961(n));
v353335 = DirInverseCorrect(vector(up_to,n,A353420(n)));
A353335(n) = v353335[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A353420(n/d) * a(d).

a(n) = A353336(n) - A353420(n).

A353420 a(n) = A126760(A003961(n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 9, 3, 5, 2, 6, 4, 12, 1, 7, 9, 8, 3, 19, 5, 10, 2, 17, 6, 42, 4, 11, 12, 13, 1, 22, 7, 26, 9, 14, 8, 29, 3, 15, 19, 16, 5, 59, 10, 18, 2, 41, 17, 32, 6, 20, 42, 31, 4, 39, 11, 21, 12, 23, 13, 92, 1, 40, 22, 24, 7, 49, 26, 25, 9, 27, 14, 82, 8, 48, 29, 28, 3, 209, 15, 30, 19, 45, 16, 52, 5, 33

Offset: 1

Antti Karttunen, Apr 20 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A003602, A003961, A048673, A126760, A249735, A249745, A249746.

Cf. A353335 (Dirichlet inverse), A353336 (sum with it).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A126760(n) = {n&&n\=3^valuation(n, 3)<A126760
A353420(n) = A126760(A003961(n));

a(n) = A126760(A003961(n)) = A249746(A003602(n)) = A126760(A249735(A003602(n))).
a(n) = A353336(4*n) = A353336(n) - A353335(n).
For all n >= 1, a(n) = a(2*n) = a(A000265(n)).
For all n >= 1, A249745(a(n)) = A003602(n).

cp's OEIS Frontend

A323894 Sum of A048673 and its Dirichlet inverse, A323893.

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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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A353335 Dirichlet inverse of A353420.

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A353420 a(n) = A126760(A003961(n)).

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