%I A072507
%S A072507 1,2,0,0,0,0,0,0,0,0,0
%N A072507 Start of n consecutive integers with n divisors, or 0 if no such number 
               exists.
%C A072507 a(3) = 0 because only perfect prime squares have three divisors.
%C A072507 Comments from T. D. Noe: "Note that a(n)=0 for odd n > 1 because a number 
               has an odd number of divisors only if it is a square and there are 
               no consecutive positive squares. Also, a(4)=0 because one of four 
               consecutive numbers would be a multiple of 4 and have 4 divisors 
               only if it is 8.
%C A072507 "Similarly, a(6)=0 because one of six consecutive number would be a multiple 
               of 6 and the only multiples of 6 having 6 divisors are 12 and 18. 
               For a(8), one of the eight consecutive numbers must be an odd multiple 
               of 4, which cannot have 8 divisors. Interestingly, the 7 consecutive 
               numbers starting at 171893 have 8 divisors.
%C A072507 "Similarly, for a(10), one of the ten consecutive numbers must be an 
               odd multiple of 4, which would have 3x divisors. It is also easy 
               to verify that a(n)=0 for n=14,16,20,22,26,28,32,34,... It seems 
               likely that a(n)=0 for n>2."
%C A072507 This sequence is zero for all but finitely many n. If k = floor(log_2(n)), 
               there must be at least one term exactly divisible by 2^j for any 
               j < k; hence the number of divisors must be divisible by j+1, or 
               more generally by lcm_{i<=k} i. The only values of n divisible by 
               this lcm are 1,2,3,4,6,12,24,60 and 120. For example, for n=30, there 
               must be an element divisible by exactly 8, so its number of divisors 
               is divisible by 4. For n = 60, there must by two numbers 8k and 8(k+2) 
               with k odd; then k and k+2 must each have 15 divisors, making them 
               squares. Together with the comments from T. D. Noe, this leaves only 
               12, 24 and 120 as open questions. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jul 14 2006
%D A072507 R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third 
               Edition, 2004, B12.
%e A072507 a(2) = 2 as 2 and 3 are the first (by chance the only) set of two consecutive 
               integers with two divisors.
%Y A072507 Cf. A000005 (number of divisors of n).
%Y A072507 Cf. A006558 (start of first run of n consecutive integers with same number 
               of divisors).
%Y A072507 Cf. A119479.
%Y A072507 Sequence in context: A122840 A083919 A063665 this_sequence A130779 A130706 
               A000038
%Y A072507 Adjacent sequences: A072504 A072505 A072506 this_sequence A072508 A072509 
               A072510
%K A072507 more,nonn
%O A072507 1,2
%A A072507 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 22 2002
%E A072507 More terms from T. D. Noe (noe(AT)sspectra.com), Dec 04 2004