|
|
A307037
|
|
The unitary analog of the alternating sum-of-divisors function (A206369).
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|