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.

A336840 Inverse Möbius transform of A048673.

Original entry on oeis.org

1, 3, 4, 8, 5, 14, 7, 22, 17, 18, 8, 42, 10, 26, 26, 63, 11, 65, 13, 55, 38, 30, 16, 124, 30, 38, 80, 81, 17, 100, 20, 185, 44, 42, 50, 206, 22, 50, 56, 164, 23, 148, 25, 94, 127, 62, 28, 368, 68, 117, 62, 120, 31, 316, 58, 244, 74, 66, 32, 318, 35, 78, 189, 550, 74, 172, 37, 133, 92, 196, 38, 626, 41, 86, 174, 159
Offset: 1

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Comments

Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).
The local minima occur on primes p, where p/2 < a(p) <= (p+1).

Links

Crossrefs

Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336840(n) = sumdiv(n,d,A048673(d));
  • PARI
    A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };

Formula

a(n) = Sum_{d|n} A048673(d).
a(n) = (1/2) * (A000005(n) + A003973(n)).
a(n) = A113415(A003961(n)). - Antti Karttunen, Jun 01 2022
a(n) = A349371(A003961(n)) = A364063(A048673(n)). - Antti Karttunen, Nov 30 2024

A364066 Expansion of Sum_{k>0} k * x^k / (1 - x^(3*k-1)).

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 10, 8, 10, 10, 15, 14, 14, 14, 20, 16, 20, 18, 28, 20, 22, 24, 30, 24, 26, 30, 40, 28, 30, 30, 40, 34, 39, 34, 48, 36, 44, 38, 50, 46, 42, 44, 58, 44, 46, 46, 74, 52, 50, 50, 68, 54, 54, 62, 70, 56, 66, 58, 82, 60, 76, 64, 80, 64, 66, 66, 97, 78, 70, 74, 90, 74, 74, 80, 114, 76, 88, 78, 100
Offset: 1

Views

Author

Seiichi Manyama, Jul 04 2023

Keywords

Links

Crossrefs

Cf. A364063, A364085.
Cf. A359211, A363890.

Programs

  • Mathematica
    a[n_] := DivisorSum[3*n - 1, # + 1 &, Mod[#, 3] == 2 &]/3; Array[a, 100] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(3*n-1, d, (d%3==2)*(d+1))/3;

Formula

a(n) = (1/3) * Sum_{d | 3*n-1, d==2 (mod 3)} (d+1).
G.f.: Sum_{k>0} x^(2*k-1) / (1 - x^(3*k-2))^2.

A364085 Expansion of Sum_{k>0} k * x^k / (1 - x^(4*k-1)).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 8, 8, 11, 11, 11, 12, 14, 17, 15, 19, 17, 18, 24, 20, 21, 23, 25, 29, 29, 26, 27, 29, 35, 32, 32, 32, 33, 46, 35, 39, 40, 38, 47, 41, 41, 42, 49, 55, 45, 47, 50, 48, 64, 50, 53, 59, 53, 65, 56, 56, 57, 64, 71, 60, 69, 67, 63, 82, 67, 66, 68, 68, 86, 79, 71, 74, 74, 89, 81, 77, 77, 78
Offset: 1

Views

Author

Seiichi Manyama, Jul 04 2023

Keywords

Crossrefs

Cf. A364063, A364066.
Cf. A078703.

Programs

  • Mathematica
    a[n_] := DivisorSum[4*n - 1, # + 1 &, Mod[#, 4] == 3 &]/4; Array[a, 100] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(4*n-1, d, (d%4==3)*(d+1))/4;

Formula

a(n) = (1/4) * Sum_{d | 4*n-1, d==3 (mod 4)} (d+1).
G.f.: Sum_{k>0} x^(3*k-2) / (1 - x^(4*k-3))^2.
Showing 1-3 of 3 results.

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