A360522 a(n) = Sum_{d|n} Max({d'; d'|n, gcd(d, d') = 1}).

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, 42, 30, 48, 30, 72, 32, 37, 48, 54, 48, 66, 38, 60, 56, 66, 42, 96, 44, 72, 66, 72, 48, 80, 51, 81, 72, 84, 54, 90, 72, 88, 80, 90, 60, 144, 62, 96, 88, 70, 84, 144, 68

Offset: 1

Author

Amiram Eldar, Feb 10 2023

Keywords

• nonn
• mult

Comments

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.

Denoted by Sur(n) in Khan (2005).

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).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Mizan R. Khan, Problem 10922, The American Mathematical Monthly, Vol. 109, No. 2 (2002), p. 201; Michael R. Avidon, The Sum of Divisors Won't Die, solution, ibid., Vol. 110, No. 10 (2003), p. 959.
• Mizan R. Khan, A variant of the divisor functions sigma_a(n), JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.

Crossrefs

Cf. A000203, A005117, A034448, A360523.

Cf. A065465, A072691, A152649, A330523.

Cf. A048691, A328485.

Programs

Mathematica

f[p_, e_] := p^e + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Formula

Multiplicative with a(p^e) = p^e + e.

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)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = A072691 * A065465 = A152649 * A330523 = 0.7250160726810604158... .

a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).

limsup_{n->oo} sigma(n)/a(n) = oo, where sigma(n) is the sum of divisors of n (A000203) (Khan, 2002).

liminf_{n->oo} a(n)/usigma(n) = 1, where usigma(n) is the sum of unitary divisors of n (A034448) (Khan, 2005).

limsup_{n->oo} a(n)/usigma(n) = (55/54) * Product_{p prime} (1 + 1/(p^2+1)) = 1.4682298236... (Khan, 2005).