A349126 -id:A349126 - cp's OEIS frontend

A349135 Sum of Kimberling's paraphrases (A003602) and its Dirichlet inverse.

2, 0, 0, 1, 0, 4, 0, 1, 4, 6, 0, 2, 0, 8, 12, 1, 0, 6, 0, 3, 16, 12, 0, 2, 9, 14, 12, 4, 0, 4, 0, 1, 24, 18, 24, 5, 0, 20, 28, 3, 0, 6, 0, 6, 26, 24, 0, 2, 16, 17, 36, 7, 0, 16, 36, 4, 40, 30, 0, 8, 0, 32, 36, 1, 42, 10, 0, 9, 48, 12, 0, 5, 0, 38, 46, 10, 48, 12, 0, 3, 37, 42, 0, 11, 54, 44, 60, 6, 0, 20, 56, 12

Offset: 1

Antti Karttunen, Nov 13 2021

sign

Question: Are all terms nonnegative?

The answer to the above question is no, because A323894 (which is a prime-shifted version of this sequence) also contains negative values. For example, for n=72747675, 88062975, 130945815, 111035925 we get here a(n) = -14126242, -17546656, -14460312, -22677277. The indices are obtained by prime-shifting with A003961 the four indices mentioned in the Apr 20 2022 comment of A323894. - Antti Karttunen, Nov 30 2024

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

Cf. A003602 (also quadrisection of this sequence), A349134, A323894 [= a(A003961(n))].

Cf. also A323882, A349126.

k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; d[1] = 1; d[n_] := d[n] = -DivisorSum[n, d[#]*k[n/#] &, # < n &]; a[n_] := k[n] + d[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

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(dA003602(n) = (1+(n>>valuation(n,2)))/2;
v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
A349134(n) = v349134[n];
A349135(n) = (A003602(n)+A349134(n));

A349135(n) = if(1==n,2,-sumdiv(n, d, if(1==d||n==d,0,A003602(d)*A349134(n/d)))); \\ (Demonstrates the "cut convolution" formula) - Antti Karttunen, Nov 13 2021

A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349135(n) = (A003602(n)+A349134(n)); \\ Antti Karttunen, Nov 30 2024

a(n) = A003602(n) + A349134(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A003602(d) * A349134(n/d).
For all n >= 1, a(4*n) = A003602(n). - Antti Karttunen, Dec 07 2021

A349125 Dirichlet inverse of A064989, 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

1, -1, -2, 0, -3, 2, -5, 0, 0, 3, -7, 0, -11, 5, 6, 0, -13, 0, -17, 0, 10, 7, -19, 0, 0, 11, 0, 0, -23, -6, -29, 0, 14, 13, 15, 0, -31, 17, 22, 0, -37, -10, -41, 0, 0, 19, -43, 0, 0, 0, 26, 0, -47, 0, 21, 0, 34, 23, -53, 0, -59, 29, 0, 0, 33, -14, -61, 0, 38, -15, -67, 0, -71, 31, 0, 0, 35, -22, -73, 0, 0, 37, -79

Offset: 1

Antti Karttunen, Nov 13 2021

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

Cf. A008683, A064989, A151800, A349126, A349127.

Cf. also A055615, A346234, A349134.

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

A064989(n) = { my(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); };
A349125(n) = (moebius(n)*A064989(n));

A349125(n) = { my(f = factor(n)); prod(i=1, #f~, if(1

from sympy import prevprime, factorint, prod
def f(p, e):
    return 0 if e > 1 else -1 if p == 2 else -prevprime(p)
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 13 2021

a(1) = 1; a(n) = -Sum_{d|n, d < n} A064989(n/d) * a(d).
a(n) = A349126(n) - A064989(n).
Multiplicative with a(p^e) = 0 if e > 1, -1 if p = 2 and -prevprime(p) otherwise. - Sebastian Karlsson, Nov 13 2021
a(n) = A008683(n) * A064989(n). [Because A064989 is fully multiplicative. See "Properties" section in the Wikipedia article]

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).

A349359 Sum of A064216 and its Dirichlet inverse, where A064216 = A064989(2n-1), and A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 12, 9, 16, 0, 22, 0, 44, 24, 5, 0, 40, 0, 60, 66, 40, 0, 14, 16, 36, 51, 10, 0, 106, 0, 82, 60, 56, 88, 26, 0, 124, 54, -10, 0, -46, 0, 144, 134, 48, 0, 235, 121, 140, 84, 86, 0, 19, 80, -108, 186, 136, 0, -44, 0, 236, 211, 29, 72, 158, 0, 216, 72, 62, 0, 152, 0, 284, 190, 10, 220, 98, 0, 260, 181

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Compare to A323894 which in contrast to this sequence seems to have only nonnegative terms.

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

Cf. A064216, A064989, A349358.

Cf. also A323894, A349126.

A064989(n) = { my(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); };
A064216(n) = A064989((2*n)-1);
memoA349358 = Map();
A349358(n) = if(1==n,1,my(v); if(mapisdefined(memoA349358,n,&v), v, v = -sumdiv(n,d,if(dA064216(n/d)*A349358(d),0)); mapput(memoA349358,n,v); (v)));
A349359(n) = (A064216(n)+A349358(n));

a(n) = A064216(n) + A349358(n).

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A064216(d) * A349358(n/d).

cp's OEIS Frontend

A349135 Sum of Kimberling's paraphrases (A003602) and its Dirichlet inverse.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A349125 Dirichlet inverse of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A349359 Sum of A064216 and its Dirichlet inverse, where A064216 = A064989(2n-1), and A064989 is fully multiplicative with a(2) = 1 and a(p) = prevprime(p) for odd primes p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula