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Numbers x such that tau(x)/tau(x-1) is an integer, where tau() is the number of divisors function.
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%I #23 Feb 02 2019 03:15:31

%S 2,3,6,8,12,14,15,18,20,22,24,27,30,32,34,35,38,39,40,42,44,45,48,50,

%T 54,56,58,60,62,66,68,70,72,74,76,78,80,84,86,87,88,90,94,95,96,98,99,

%U 102,104,105,108,110,114,117,119,120,123,126,128,130,132,134,135,136,138

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

%C 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.

%H Jinyuan Wang, <a href="/A086552/b086552.txt">Table of n, a(n) for n = 1..10000</a>

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

%p 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

%o (PARI) isok(n) = denominator(numdiv(n)/numdiv(n-1)) == 1; \\ _Michel Marcus_, Apr 12 2018

%Y Cf. A057922, A086550, A086551.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Aug 28 2003

%E More terms from _David Wasserman_ and _Emeric Deutsch_, Mar 25 2005