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a(n) = k: k is smallest integer > 1 such that sign(d(1)-d(2)) = sign(d(k)-d(k+1)), sign(d(2)-d(3)) = sign(d(k+1)-d(k+2)),...,sign(d(n)-d(n+1)) = sign(d(k+n-1)-d(k+n)), where sign is (-,0,+) and d(m) = the number of positive divisors of m.
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%I #8 Oct 03 2015 23:06:56

%S 3,13,13,13,13,13,13,13,121,121,121,121,985,10489,10489,10489,10489,

%T 10489,10489,10489,10489,10489,10489,10489,10489,10489,10489,10489,

%U 10489,10489,10489,10489

%N a(n) = k: k is smallest integer > 1 such that sign(d(1)-d(2)) = sign(d(k)-d(k+1)), sign(d(2)-d(3)) = sign(d(k+1)-d(k+2)),...,sign(d(n)-d(n+1)) = sign(d(k+n-1)-d(k+n)), where sign is (-,0,+) and d(m) = the number of positive divisors of m.

%C a(33) > 2300000 if it exists. - _R. J. Mathar_, Apr 21 2008

%p A000005 := proc(n) numtheory[tau](n) ; end: A137947 := proc(n) local k,o,works ; for k from 2 do works := true ; for o from 1 to n do if signum(A000005(o)-A000005(o+1)) <> signum(A000005(k+o-1)-A000005(k+o)) then works := false ; break ; fi ; od: if works then RETURN(k) ; fi ; od: end: for n from 1 do print(n,A137947(n)) ; od: # _R. J. Mathar_, Apr 21 2008

%Y Cf. A000005.

%K nonn

%O 1,1

%A _Leroy Quet_, Feb 24 2008, Mar 09 2008

%E More terms from _R. J. Mathar_, Apr 21 2008