A349349 -id:A349349 - 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

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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

A349126 Sum of A064989 and its Dirichlet inverse, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 6, 0, 2, 0, 10, 12, 1, 0, 4, 0, 3, 20, 14, 0, 2, 9, 22, 8, 5, 0, 0, 0, 1, 28, 26, 30, 4, 0, 34, 44, 3, 0, 0, 0, 7, 12, 38, 0, 2, 25, 9, 52, 11, 0, 8, 42, 5, 68, 46, 0, 6, 0, 58, 20, 1, 66, 0, 0, 13, 76, 0, 0, 4, 0, 62, 18, 17, 70, 0, 0, 3, 16, 74, 0, 10, 78, 82, 92, 7, 0, 12, 110, 19, 116

Offset: 1

Antti Karttunen, Nov 13 2021

Question: Are all terms nonnegative?

Answer: All terms certainly are >= 0. See Sebastian Karlsson's Nov 13 2021 multiplicative formula for A349125. - 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

Cf. A001248, A064989, A030059, A030229, A280076, A349125.
Cf. also A322581, A349135.
Coincides with A349349 on odd numbers.

f1[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a1[1] = 1; a1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := If[e == 1, If[p == 2, -1, -NextPrime[p, -1]], 0]; a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := a1[n] + a2[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

A349126(n) = (A064989(n)+A349125(n)); \\ Needs also code from A349125.

A349126(n) = if(1==n,2,-sumdiv(n, d, if(1==d||n==d,0,A064989(d)*A349125(n/d)))); \\ (This demonstrates the "cut convolution" formula) - Antti Karttunen, Nov 13 2021

a(n) = A064989(n) + A349125(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A064989(d) * A349125(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A064989(A030229(n)).
For all n >= 1, a(A001248(n)) = A280076(n).

A349348 Dirichlet inverse of A252463, 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

1, -1, -2, -1, -3, 1, -5, -1, 0, 1, -7, 2, -11, 3, 6, -1, -13, -1, -17, 3, 10, 3, -19, 3, 0, 9, 0, 5, -23, -1, -29, -1, 14, 9, 15, 1, -31, 15, 22, 5, -37, -3, -41, 7, 0, 15, -43, 4, 0, -4, 26, 11, -47, -3, 21, 7, 34, 17, -53, -2, -59, 27, 0, -1, 33, -3, -61, 13, 38, -3, -67, 2, -71, 25, 0, 17, 35, -9, -73, 7, 0, 33

Offset: 1

Antti Karttunen, Nov 15 2021

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

Coincides with A349125 on odd numbers.
Cf. A064989, A252463, A349349.
Cf. also A348045, A349437, A349438.

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];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A252463(n/d) * a(d).
a(n) = A349349(n) - A252463(n).
For all n >= 1, a(2n-1) = A349125(2n-1).

cp's OEIS Frontend

A323894 Sum of A048673 and its Dirichlet inverse, A323893.

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A349126 Sum of A064989 and its Dirichlet inverse, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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A349348 Dirichlet inverse of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

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Crossrefs

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PARI

Formula