A332882 -id:A332882 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Feb 28 2020

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

			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, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

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

a:= n-> numer(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

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

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);

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)

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

Amiram Eldar, Jul 20 2024

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, ... .

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002110, A005117, A034444, A034448, A077610, A306633, A332882, A332883, A343525, A374784 (denominators).

Similar sequences: A374777/A374778, A374786/A374787.

f[p_, e_] := 1 + 1/(2*p^e); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])));}

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).

A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 3, 19, 2, 21, 11, 23, 2, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 35, 18, 37, 19, 39, 20, 41, 7, 43, 11, 3, 23, 47, 12, 49, 25, 17, 26, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Denominator of sum of reciprocals of unitary divisors of n.

			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, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A007947, A017666, A034448, A077610, A319677, A323166, A327158 (positions of 1's), A332881, A332882 (numerators).

a:= n-> denom(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Denominator

a(n) = denominator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = denominator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = denominator of usigma(n)/n.
a(p) = p, where p is prime.
a(n) = n / A323166(n). - Antti Karttunen, Nov 13 2021

cp's OEIS Frontend

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

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

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A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

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Comments

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Crossrefs

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Maple

Mathematica

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Formula