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 1-4 of 4 results.

A349567 Dirichlet convolution of A133494 [3^(n-1)] with A349452 (Dirichlet inverse of A011782, 2^(n-1)).

Original entry on oeis.org

1, 1, 5, 17, 65, 197, 665, 2017, 6285, 19025, 58025, 174565, 527345, 1584737, 4766245, 14311841, 42981185, 128995317, 387158345, 1161697825, 3485732845, 10458138977, 31376865305, 94134428213, 282412758225, 847253996225, 2541798693045, 7625460083185, 22876524019505, 68629830861205, 205890058352825, 617671220125537
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Comments

Dirichlet convolution of this sequence with A034738 produces A034754.

Links

Crossrefs

Cf. A011782, A133494, A349452, A349568 (Dirichlet inverse).
Cf. also A034738, A034754, A349563, A349565, A349569.

Programs

  • Mathematica
    s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * 2^(n/# - 1) &, # < n &]; a[n_] := DivisorSum[n, 3^(# - 1) * s[n/#] &]; Array[a, 32] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A011782(n) = (2^(n-1));
memoA349452 = Map();
A349452(n) = if(1==n,1,my(v); if(mapisdefined(memoA349452,n,&v), v, v = -sumdiv(n,d,if(dA011782(n/d)*A349452(d),0)); mapput(memoA349452,n,v); (v)));
A349567(n) = sumdiv(n,d,(3^(d-1)) * A349452(n/d));

Formula

a(n) = Sum_{d|n} 3^(d-1) * A349452(n/d).

A349453 Dirichlet inverse of A133494, 3^(n-1).

Original entry on oeis.org

1, -3, -9, -18, -81, -189, -729, -2052, -6480, -19197, -59049, -175446, -531441, -1589949, -4781511, -14335704, -43046721, -129097152, -387420489, -1162141182, -3486771279, -10459998909, -31381059609, -94142073420, -282429529920, -847285420797, -2541865710960, -7625587899366, -22876792454961, -68630348286531
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Links

Crossrefs

Cf. A133494.
Cf. also A349449, A349450, A349451, A349452, A349568.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * 3^(n/# - 1) &, # < n &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A133494(n) = max(1, 3^(n-1));
memoA349453 = Map();
A349453(n) = if(1==n,1,my(v); if(mapisdefined(memoA349453,n,&v), v, v = -sumdiv(n,d,if(dA133494(n/d)*A349453(d),0)); mapput(memoA349453,n,v); (v)));

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A133494(n/d) * a(d).
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} 3^(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022

A349564 Dirichlet convolution of A011782 [2^(n-1)] with A349450 [Dirichlet inverse of right-shifted Catalan numbers].

Original entry on oeis.org

1, 1, 2, 2, 2, -14, -68, -308, -1178, -4366, -15772, -56780, -203916, -734772, -2658088, -9662208, -35292134, -129514026, -477376556, -1766739436, -6563071972, -24464170892, -91478369336, -343051227304, -1289887370136, -4861912851116, -18367285963792, -69533416706328, -263747683314904, -1002241679797688
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Comments

Dirichlet convolution with A034731 gives A034729.

Links

Crossrefs

Cf. A011782, A349450.
Cf. A000108, A011782, A349452, A349563 (Dirichlet inverse).
Cf. also A034729, A034731, A349566, A349568.

Programs

  • Mathematica
    s[1] = 1; s[n_] := s[n] = -DivisorSum[n, s[#] * CatalanNumber[n/# - 1] &, # < n &]; a[n_] := DivisorSum[n, 2^(# - 1) * s[n/#] &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A000108(n) = (binomial(2*n, n)/(n+1));
memoA349450 = Map();
A349450(n) = if(1==n,1,my(v); if(mapisdefined(memoA349450,n,&v), v, v = -sumdiv(n,d,if(dA000108((n/d)-1)*A349450(d),0)); mapput(memoA349450,n,v); (v)));
A349564(n) = sumdiv(n,d,2^(d-1)*A349450(n/d));

Formula

a(n) = Sum_{d|n} 2^(d-1) * A349450(n/d).

A349570 Dirichlet convolution of A011782 [2^(n-1)] with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 1, 4, 11, 24, 57, 112, 244, 480, 1013, 1972, 4083, 8064, 16331, 32512, 65519, 130488, 262125, 523244, 1048377, 2095104, 4194281, 8384176, 16777136, 33546240, 67108096, 134201316, 268435427, 536836584, 1073741793, 2147418112, 4294964213, 8589803520, 17179868787, 34359470272, 68719476699, 137438429184, 274877894643
Offset: 1

Views

Author

Antti Karttunen, Nov 22 2021

Keywords

Comments

Dirichlet convolution of this sequence with phi (A000010) is A000740, with sigma (A000203) it is A034729, and with A018804 it is A034738.

Links

Crossrefs

Cf. A011782, A055615, A349569 (Dirichlet inverse).
Cf. also A000010, A000740, A000203, A018804, A034729, A034738, A349564, A349566, A349568.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, # * MoebiusMu[#] * 2^(n/# - 1) &]; Array[a, 40] (* Amiram Eldar, Nov 22 2021 *)
  • PARI
    A055615(n) = (n*moebius(n));
A349570(n) = sumdiv(n,d,(2^(d-1)) * A055615(n/d));

Formula

a(n) = Sum_{d|n} 2^(d-1) * A055615(n/d).
Showing 1-4 of 4 results.

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

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