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

1,3

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".

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A008836, A034448, A206369.

Sequence in context: A055127 A152669 A324051 * A049060 A092462 A256357

Adjacent sequences: A307034 A307035 A307036 * A307038 A307039 A307040

Keyword

nonn,mult

Author

Amiram Eldar, Mar 21 2019

Status

approved