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.
%I A374783 #13 Jul 20 2024 14:41:10
%S A374783 1,5,7,9,11,35,15,17,19,11,23,21,27,75,77,33,35,95,39,99,5,115,47,119,
%T A374783 51,135,55,135,59,77,63,65,161,175,33,19,75,195,63,187,83,25,87,207,
%U A374783 209,235,95,77,99,51,245,243,107,275,23,255,91,295,119,231,123,315
%N A374783 Numerator of the mean unitary abundancy index of the unitary divisors of n.
%C A374783 The unitary abundancy index of a number k is A034448(k)/k = A332882(k)/A332883(k).
%C A374783 The record values of a(n)/A374784(n) are attained at the primorial numbers (A002110).
%C A374783 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, ... .
%H A374783 Amiram Eldar, <a href="/A374783/b374783.txt">Table of n, a(n) for n = 1..10000</a>
%F A374783 Let f(n) = a(n)/A374784(n). Then:
%F A374783 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 A374783 f(n) is multiplicative with f(p^e) = 1 + 1/(2*p^e).
%F A374783 f(n) = (Sum_{d|n, gcd(d, n/d) = 1} d*ud(d))/(n*ud(n)) = A343525(n)/(n*A034444(n)).
%F A374783 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))).
%F A374783 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).
%F A374783 Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
%F A374783 f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).
%e A374783 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.
%t A374783 f[p_, e_] := 1 + 1/(2*p^e); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
%o A374783 (PARI) a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])));}
%Y A374783 Cf. A002110, A005117, A034444, A034448, A077610, A306633, A332882, A332883, A343525, A374784 (denominators).
%Y A374783 Similar sequences: A374777/A374778, A374786/A374787.
%K A374783 nonn,easy,frac
%O A374783 1,2
%A A374783 _Amiram Eldar_, Jul 20 2024