OFFSET
1,2
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).
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.
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... .
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).
MATHEMATICA
f[p_, e_] := p^e + e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] + f[i, 2]); }
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, Feb 10 2023
STATUS
approved