%I #12 Feb 11 2023 07:56:54
%S 1,3,4,6,6,12,8,11,11,18,12,24,14,24,24,20,18,33,20,36,32,36,24,44,27,
%T 42,30,48,30,72,32,37,48,54,48,66,38,60,56,66,42,96,44,72,66,72,48,80,
%U 51,81,72,84,54,90,72,88,80,90,60,144,62,96,88,70,84,144,68
%N a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).
%C a(n) is the sum of delta_d(n) over the divisors d of n, where delta_d(n) is the greatest divisor of n that is relatively prime to n.
%C Denoted by Sur(n) in Khan (2005).
%C Related sequences: A048691(n) = Sum_{d|n} #{d'; d' | n, gcd(d, d') = 1}, and A328485(n) = Sum_{d|n} Sum_{d' | n, gcd(d, d') = 1} d' (number and sum of divisors instead of maximal divisor, respectively).
%H Amiram Eldar, <a href="/A360522/b360522.txt">Table of n, a(n) for n = 1..10000</a>
%H Mizan R. Khan, <a href="http://www.jstor.org/stable/2695336">Problem 10922</a>, The American Mathematical Monthly, Vol. 109, No. 2 (2002), p. 201; Michael R. Avidon, <a href="http://www.jstor.org/stable/3647978">The Sum of Divisors Won't Die</a>, solution, ibid., Vol. 110, No. 10 (2003), p. 959.
%H Mizan R. Khan, <a href="http://www.pphmj.com/abstract/699.htm">A variant of the divisor functions sigma_a(n)</a>, JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.
%F Multiplicative with a(p^e) = p^e + e.
%F Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(2*s)).
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691 * A065465 = A152649 * A330523 = 0.7250160726810604158... .
%F a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
%F limsup_{n->oo} sigma(n)/a(n) = oo, where sigma(n) is the sum of divisors of n (A000203) (Khan, 2002).
%F liminf_{n->oo} a(n)/usigma(n) = 1, where usigma(n) is the sum of unitary divisors of n (A034448) (Khan, 2005).
%F limsup_{n->oo} a(n)/usigma(n) = (55/54) * Product_{p prime} (1 + 1/(p^2+1)) = 1.4682298236... (Khan, 2005).
%t f[p_, e_] := p^e + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]);}
%Y Cf. A000203, A005117, A034448, A360523.
%Y Cf. A065465, A072691, A152649, A330523.
%Y Cf. A048691, A328485.
%K nonn,mult
%O 1,2
%A _Amiram Eldar_, Feb 10 2023