cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A349439 a(n) = A349437(n) + A349438(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, 1, 1, 4, 0, -1, 0, 4, 4, 1, 0, 3, 0, -2, 4, 8, 0, -2, 4, 4, 5, -2, 0, -8, 0, 1, 8, 8, 8, -4, 0, 4, 4, -4, 0, -8, 0, -4, 10, 8, 0, -3, 4, 8, 8, -2, 0, 2, 16, -4, 4, 12, 0, -6, 0, 4, 10, 1, 8, -16, 0, -4, 8, -16, 0, -6, 0, 12, 16, -2, 16, -8, 0, -6, 19, 8, 0, -6, 16, 4, 12, -8, 0, -24, 8, -4, 4
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Links

Crossrefs

Cf. A349437, A349438.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] + # * sinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    A349439(n) = (A349437(n)+A349438(n)); \\ Needs also code from A349437 and A349438.

Formula

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A349437(d) * A349438(n/d). [As the sequences are Dirichlet inverses of each other]

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

Views

Author

Antti Karttunen, Nov 15 2021

Keywords

Links

Crossrefs

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

Programs

  • PARI
    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];

Formula

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).

A349438 Dirichlet convolution of A000027 with 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, 1, 1, 2, 0, 2, 1, 3, 0, 4, -1, 2, 0, 2, 1, 4, -1, 2, -2, 2, 0, 4, -2, 10, 0, 9, -2, 6, -4, 2, 1, 4, 0, 4, -4, 6, 0, 2, -4, 4, -4, 2, -4, 6, 0, 4, -3, 14, -4, 4, -2, 6, -6, 8, -4, 2, 0, 6, -6, 2, 0, 6, 1, 4, -8, 6, -4, 4, -8, 4, -6, 2, 0, 10, -2, 8, -4, 6, -6, 27, 0, 4, -6, 8, 0, 6, -8, 6, -16, 4, -4, 2, 0, 4
Offset: 1

Views

Author

Antti Karttunen, Nov 18 2021

Keywords

Comments

Convolving this sequence with A348045 gives Euler phi, A000010.
It might first seem that A000265(a(p^k)) = p^(k-1) for all odd primes and all exponents k >= 1, but this does not hold for prime 37. However, with p=37, identity A065330(A349438(37^k)) = 37^(k-1) seems to hold for all exponents k >= 1. - Antti Karttunen, Nov 20 2021

Links

Crossrefs

Cf. A000027, A064989, A252463, A349348, A349437 (Dirichlet inverse), A349439 (sum with it).
Cf. also A000010, A000265, A065330, A348045.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    A064989(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));
memoA349348 = Map();
A349348(n) = if(1==n,1,my(v); if(mapisdefined(memoA349348,n,&v), v, v = -sumdiv(n,d,if(dA252463(n/d)*A349348(d),0)); mapput(memoA349348,n,v); (v)));
A349438(n) = sumdiv(n,d,d*A349348(n/d));

Formula

a(n) = Sum_{d|n} d * A349348(n/d).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.