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.

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

A374783 Numerator of the mean unitary abundancy index of the unitary divisors of n.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 17, 19, 11, 23, 21, 27, 75, 77, 33, 35, 95, 39, 99, 5, 115, 47, 119, 51, 135, 55, 135, 59, 77, 63, 65, 161, 175, 33, 19, 75, 195, 63, 187, 83, 25, 87, 207, 209, 235, 95, 77, 99, 51, 245, 243, 107, 275, 23, 255, 91, 295, 119, 231, 123, 315
Offset: 1

Views

Author

Amiram Eldar, Jul 20 2024

Keywords

Comments

The unitary abundancy index of a number k is A034448(k)/k = A332882(k)/A332883(k).
The record values of a(n)/A374784(n) are attained at the primorial numbers (A002110).
The least number k such that a(k)/A374784(k) is larger than 2, 3, 4, ..., is A002110(9) = 223092870, A002110(314) = 7.488... * 10^878, A002110(65599) = 5.373... * 10^356774, ... .

Examples

			For n = 4, 4 has 2 unitary divisors, 1 and 4. Their unitary abundancy indices are usigma(1)/1 = 1 and usigma(4)/4 = 5/4, and their mean unitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = numerator(9/8) = 9.

Links

Crossrefs

Cf. A002110, A005117, A034444, A034448, A077610, A306633, A332882, A332883, A343525, A374784 (denominators).
Similar sequences: A374777/A374778, A374786/A374787.

Programs

  • Mathematica
    f[p_, e_] := 1 + 1/(2*p^e); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])));}

Formula

Let f(n) = a(n)/A374784(n). Then:
f(n) = (Sum_{d|n, gcd(d, n/d) = 1} usigma(d)/d) / ud(n), where usigma(n) is the sum of unitary divisors of n (A034448), and ud(n) is their number (A034444).
f(n) is multiplicative with f(p^e) = 1 + 1/(2*p^e).
f(n) = (Sum_{d|n, gcd(d, n/d) = 1} d*ud(d))/(n*ud(n)) = A343525(n)/(n*A034444(n)).
Dirichlet g.f. of f(n): zeta(s) * zeta(s+1) * Product_{p prime} (1 - 1/(2*p^(s+1)) - 1/(2*p^(2*s+1))).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} (1 + 1/(2*p*(p+1))) = 1.17443669198552182119... . For comparison, the asymptotic mean of the unitary abundancy index over all the positive integers is zeta(2)/zeta(3) = 1.368432... (A306633).
Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).

A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 3, 4, 5, 6, 2, 8, 9, 10, 9, 12, 5, 14, 12, 8, 17, 18, 5, 20, 3, 32, 18, 24, 3, 26, 21, 28, 10, 30, 12, 32, 33, 16, 27, 48, 25, 38, 30, 56, 27, 42, 16, 44, 15, 4, 36, 48, 17, 50, 39, 24, 35, 54, 14, 72, 9, 80, 45, 60, 2, 62, 48, 80, 65, 84, 24, 68, 45, 32, 72
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

Comments

Numerator of sum of reciprocals of unitary divisors of n.

Examples

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...

Links

Crossrefs

Cf. A017665, A028235, A028236, A034448, A077610, A306633, A319676, A323166, A332880, A332883 (denominators).
Cf. also A348734, A348735.

Programs

  • Maple
    a:= n-> numer(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020
  • Mathematica
    Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Numerator
  • PARI
    a(n) = numerator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

Formula

a(n) = numerator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = numerator of usigma(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A034448(n) / A323166(n). - Antti Karttunen, Nov 13 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332883(k) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Nov 21 2022
Showing 1-3 of 3 results.

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