A367988 The sum of the divisors of the square root of the largest unitary divisor of n that is a square.

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 7, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 7, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 15, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 7, 13, 1, 1, 3, 1, 1

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = (p^(e/2+1)-1)/(p-1) if e is even and 1 otherwise.

a(n) = A000203(A071974(n)).

a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).

Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1)).

Mathematica

f[p_, e_] := If[EvenQ[e], (p^(e/2 + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

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

Crossrefs

Cf. A000203, A071974, A268335, A351568, A367987.

Sequence in context: A122947 A367990 A360162 * A372692 A363925 A231147

Adjacent sequences: A367985 A367986 A367987 * A367989 A367990 A367991

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 07 2023

Status

approved