A335385 The number of tri-unitary divisors of n.

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset

1,2

Comments

A divisor d of k is tri-unitary if the greatest common bi-unitary divisor of d and k/d is 1.

Differs from A037445 at n = 32, 96, 128, 160, 224, ...

Links

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

Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.

Pentti Haukkanen, On the k-ary convolution of arithmetical functions, The Fibonacci Quarterly, Vol. 38, No. 5 (2000), pp. 440-445.

Formula

Multiplicative with a(p^e) = 4 if e = 3 or 6, and a(p^e) = 2 otherwise.

Example

a(4) = 2 since 4 has 2 tri-unitary divisors, 1 and 4. 2 is not a tri-unitary divisor of 4 since the greatest common bi-unitary divisor of 2 and 4/2 = 2 is 2 and not 1.

Mathematica

f[p_, e_] := If[e == 3 || e == 6, 4, 2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> if(x == 3 || x == 6, 4, 2), factor(n)[, 2])); \\ Amiram Eldar, Dec 18 2023

Crossrefs

Cf. A222266, A324706, A324707, A324708, A324709.

Cf. A000005, A034444, A037445, A048105, A049419, A286324, A322483.

Sequence in context: A084718 A154851 A281854 * A037445 A318307 A331109

Adjacent sequences: A335382 A335383 A335384 * A335386 A335387 A335388

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 04 2020

Status

approved