A086552 Numbers x such that tau(x)/tau(x-1) is an integer, where tau() is the number of divisors function.

2, 3, 6, 8, 12, 14, 15, 18, 20, 22, 24, 27, 30, 32, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 84, 86, 87, 88, 90, 94, 95, 96, 98, 99, 102, 104, 105, 108, 110, 114, 117, 119, 120, 123, 126, 128, 130, 132, 134, 135, 136, 138

Offset: 1

Author

Amarnath Murthy, Aug 28 2003

Keywords

• nonn

Comments

Conjecture: (1) tau(x)/tau(x-1) = n has solutions for every n. (2) If x is the smallest number for a given n such that tau(x)/tau(x-1) = n > 1, then x-1 is a prime.

Examples

			12 is a member as tau(12)/tau(11) = 3, 15 is a member as tau(15)/tau(14) = 1.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A057922, A086550, A086551.

Programs

Maple

with(numtheory): a:=proc(n) if type(tau(n)/tau(n-1), integer)=true then n else fi end: seq(a(n),n=2..150); # Emeric Deutsch, Mar 25 2005

PARI

isok(n) = denominator(numdiv(n)/numdiv(n-1)) == 1; \\ Michel Marcus, Apr 12 2018

Extensions

More terms from David Wasserman and Emeric Deutsch, Mar 25 2005