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

Author

Amiram Eldar, Jun 04 2020

Keywords

• nonn
• easy
• mult

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, ...

Examples

			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.
		

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.

Crossrefs

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

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

Programs

Mathematica

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

PARI

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

Formula

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