A307037 The unitary analog of the alternating sum-of-divisors function (A206369).

1, 1, 2, 5, 4, 2, 6, 7, 10, 4, 10, 10, 12, 6, 8, 17, 16, 10, 18, 20, 12, 10, 22, 14, 26, 12, 26, 30, 28, 8, 30, 31, 20, 16, 24, 50, 36, 18, 24, 28, 40, 12, 42, 50, 40, 22, 46, 34, 50, 26, 32, 60, 52, 26, 40, 42, 36, 28, 58, 40, 60, 30, 60, 65, 48, 20, 66, 80

Offset: 1

Author

Amiram Eldar, Mar 21 2019

Keywords

• nonn
• mult

Comments

Also equals to the number of integers 1 <= k <= n such that the largest divisor of k which is a unitary divisor of n, gcd(k, n)*, is a square.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• László Tóth, A survey of the alternating sum-of-divisors function, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107, section 6, "Unitary analog".

Crossrefs

Cf. A008836, A034448, A206369.

Programs

Mathematica

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

PARI

a(n) = my(f=factor(n)); for (k=1, #f~, e=f[k,2]; f[k,1] = f[k,1]^e+(-1)^e; f[k,2]=1); factorback(f); \\ Michel Marcus, Mar 23 2019

Formula

Multiplicative with a(p^e) = p^e + (-1)^e.

Equals Sum_{d||n} d*lambda(n/d), where the sum is over the unitary divisors of n, and lambda(n) is Liouville's function (A008836).

Sum_{k=1..n} a(k) ~ c*n^2 where c = 63*zeta(3)/(2*Pi^4) = 0.388719...

Dirichlet g.f. is zeta(s-1)*zeta(2*s)*zeta(2*s-1) / (zeta(s)*zeta(4*s-2)) (see L. Tóth, A survey ..., p. 101). - Werner Schulte, Feb 07 2021