A281188 - OEIS

%I #103 Feb 09 2017 02:40:31

%S 1,1,1,1,1,2,1,0,0,2,1,0,1,2,2,0,1,0,1,0,2,2,1,0,0,2,0,0,1,6,1,0,2,2,

%T 2,0,1,2,2,0,1,6,1,0,0,2,1,0,0,0,2,0,1,0,2,0,2,2,1,0,1,2,0,0,2,6,1,0,

%U 2,6,1,0,1,2,0,0,2,6,1,0,0,2,1,0,2,2,2,0,1,0,2,0,2,2,2,0,1,0,0,0

%N Number of refactorable numbers m such that tau(m) = n, or 0 if there are infinitely many such numbers.

%C An integer n is a refactorable number if and only if tau(n) (A000005) divides n.

%C Every number is tau(m) for some refactorable m.

%C If n is squarefree with k prime divisors, then a(n) = k! (for a proof, see the Links entry from the author).

%C Conjecture: a(n) is nonzero if and only if n is squarefree or n = 4. [This conjecture is true; see Links for a proof. - _Jon E. Schoenfield_ and _Altug Alkan_, Jan 17 2017]

%C See also Theorem 5 for the proof of conjecture in Colton link. - _Altug Alkan_, Jan 20 2017

%H Altug Alkan, <a href="/A281188/b281188.txt">Table of n, a(n) for n = 1..10000</a>

%H Franklin T. Adams-Watters, <a href="/A281188/a281188.txt">Refactorable numbers with tau squarefree</a>

%H S. Colton, <a href="https://cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2, 1999.

%H Jon E. Schoenfield and Altug Alkan, <a href="/A281188/a281188_1.txt">Refactorable numbers with tau nonsquarefree</a>

%e If n is prime, the only refactorable number m with tau(m) = n is n^(n-1), so a(n) = 1 for n prime.

%e Any number n of the form 8p, p a prime not equal to 2, has tau(n) = 8, and thus n is refactorable. Hence a(8) = 0.

%t k = 1; t[_] = 0; t[4] = 1; While[k < 100000001, m = DivisorSigma[0, k]; If[ Mod[k, m] == 0 && SquareFreeQ@ m, t[m]++]; k++]; t@# & /@ Range@20 (* _Robert G. Wilson v_, Jan 16 2017 *)

%o (PARI) a(n) = if(n==4, 1, if(issquarefree(n) == 1, omega(n)!, 0)); \\ _Altug Alkan_, Jan 18 2017

%Y Cf. A000005, A033950, A039819.

%K nonn,easy

%O 1,6

%A _Franklin T. Adams-Watters_, Jan 16 2017

%E More terms from _Altug Alkan_, Jan 17 2017