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.

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

Original entry on oeis.org

1, 3, 4, 3, 6, 2, 8, 3, 4, 9, 12, 2, 14, 12, 8, 3, 18, 2, 20, 9, 32, 18, 24, 2, 6, 21, 4, 12, 30, 12, 32, 3, 16, 27, 48, 2, 38, 30, 56, 9, 42, 16, 44, 18, 8, 36, 48, 2, 8, 9, 24, 21, 54, 2, 72, 12, 80, 45, 60, 12, 62, 48, 32, 3, 84, 24, 68, 27, 32, 72
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

Comments

Numerator of sum of reciprocals of squarefree divisors of n.
(6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - Amiram Eldar, Nov 24 2022

Examples

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

Links

Crossrefs

Cf. A001615, A008683, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.
Cf. also A348734, A348735.
Cf. A059956, A065463, A082020, A094387, A339076.

Programs

  • Maple
    a:= n-> numer(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020
  • Mathematica
    Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Numerator
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332880(n) = numerator(A001615(n)/n);

Formula

Numerators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = numerator of Sum_{d|n} mu(d)^2/d.
a(n) = numerator of psi(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A001615(n) / A306695(n) = A001615(n) / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021
From Amiram Eldar, Nov 24 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)

A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 2, 17, 1, 19, 5, 21, 11, 23, 1, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 5, 23, 47, 1, 7, 5, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 35
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

Comments

Denominator of sum of reciprocals of squarefree divisors of n.

Examples

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

Links

Crossrefs

Cf. A001615, A008683, A017666, A048250, A007947, A109395, A187778 (positions of 1's), A306695, A308443, A308462, A332880 (numerators), A332883.

Programs

  • Maple
    a:= n-> denom(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020
  • Mathematica
    Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Denominator
  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332881(n) = denominator(A001615(n)/n);

Formula

Denominators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = denominator of Sum_{d|n} mu(d)^2/d.
a(n) = denominator of psi(n)/n.
a(p) = p, where p is prime.
a(n) = n / A306695(n) = n / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021

A377154 Expansion of e.g.f. exp(Sum_{k>=1} A000082(k)*x^k/k).

Original entry on oeis.org

1, 1, 7, 43, 385, 3721, 47911, 612067, 9559873, 157478545, 2910837511, 56866891291, 1224263236417, 27618866777113, 673173639519655, 17237263465417171, 469017851840595841, 13367670808113197857, 401964392506370969863, 12604372518766870306315, 414278024498330114803201
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 31 2024

Keywords

Links

Crossrefs

Cf. A000082, A001615, A308462.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sum[k * Product[1 + 1/p, {p, Select[Divisors[k], PrimeQ]}] * x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

Formula

a(n) ~ n! * 5^(1/6) * exp(-1/12 - 1/(20*Pi^2) - 3^(2/3)*n^(1/3) / (10^(1/3)*Pi^(4/3)) + 3^(4/3)*5^(1/3)*n^(2/3) / (2*Pi)^(2/3)) / (6^(1/3) * Pi^(5/6) * n^(2/3)).
a(n) ~ 10^(1/6) * exp(-1/12 - 1/(20*Pi^2) - 3^(2/3)*n^(1/3) / (10^(1/3)*Pi^(4/3)) + 3^(4/3)*5^(1/3)*n^(2/3) / (2*Pi)^(2/3)) * n^(n - 1/6) / (3^(1/3) * Pi^(1/3) * exp(n)).
E.g.f.: exp(Sum_{k>=1} A001615(k)*x^k).
Showing 1-3 of 3 results.

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