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A115186 Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).

%I A115186 #24 Oct 02 2024 12:31:54
%S A115186 2,9,27,135,944,5264,29888,50624,203391,3290624,6082047,32535999,
%T A115186 326481920,3274208000,6929459199,72523096064,37694578688,471672487935,
%U A115186 11557226700800,54386217385983,50624737509375,275892612890624,4870020829413375,68091093855502335,2280241934368767,809386931759611904,519017301463269375
%N A115186 Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).
%C A115186 A001222(a(n)) = A001222(a(n)+1) = n: subsequence of A045920.
%C A115186 a(16) > 4*10^10. - _Martin Fuller_, Jan 17 2006
%C A115186 a(n) <= A093548(n) <= A052215(n). - _Zak Seidov_, Jan 16 2015
%C A115186 Apparently, 4*a(n)+2 is the least number k such that k-2 and k+2 have exactly n+2 prime factors, counted with multiplicity. - _Hugo Pfoertner_, Apr 02 2024
%D A115186 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 135, p. 46, Ellipses, Paris 2008.
%e A115186 a(10) = 3290624 = 6427 * 2^9, 3290624+1 = 13 * 5^5 * 3^4:
%e A115186 A001222(3290624) = A001222(3290625) = 10.
%p A115186 f:= proc(n) uses priqueue; local t,x,p,i;
%p A115186     initialize(pq);
%p A115186     insert([-3^n, 3$n], pq);
%p A115186     do
%p A115186       t:= extract(pq);
%p A115186       x:= -t[1];
%p A115186       if numtheory:-bigomega(x-1)=n then return x-1
%p A115186       elif numtheory:-bigomega(x+1)=n then return x
%p A115186       fi;
%p A115186       p:= nextprime(t[-1]);
%p A115186       for i from n+1 to 2 by -1 while t[i] = t[-1] do
%p A115186         insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
%p A115186       od;
%p A115186     od
%p A115186 end proc:
%p A115186 seq(f(i),i=1..27); # _Robert Israel_, Sep 30 2024
%Y A115186 Cf. A001222, A045920, A052215, A093548.
%Y A115186 Equivalent sequences for longer runs: A113752 (3), A356893 (4).
%K A115186 nonn
%O A115186 1,1
%A A115186 _Reinhard Zumkeller_, Jan 16 2006
%E A115186 a(13)-a(15) from _Martin Fuller_, Jan 17 2006
%E A115186 a(16)-a(17) from _Donovan Johnson_, Apr 08 2008
%E A115186 a(18)-a(22) from _Donovan Johnson_, Jan 21 2009
%E A115186 a(23)-a(25) from _Donovan Johnson_, May 25 2013
%E A115186 a(26)-a(27) from _Robert Israel_, Sep 30 2024