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

2, 9, 27, 135, 944, 5264, 29888, 50624, 203391, 3290624, 6082047, 32535999, 326481920, 3274208000, 6929459199, 72523096064, 37694578688, 471672487935, 11557226700800, 54386217385983, 50624737509375, 275892612890624, 4870020829413375, 68091093855502335, 2280241934368767, 809386931759611904, 519017301463269375

Offset: 1

Author

Reinhard Zumkeller, Jan 16 2006

Keywords

• nonn

Comments

A001222(a(n)) = A001222(a(n)+1) = n: subsequence of A045920.

a(16) > 4*10^10. - Martin Fuller, Jan 17 2006

a(n) <= A093548(n) <= A052215(n). - Zak Seidov, Jan 16 2015

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

Examples

			a(10) = 3290624 = 6427 * 2^9, 3290624+1 = 13 * 5^5 * 3^4:
A001222(3290624) = A001222(3290625) = 10.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 135, p. 46, Ellipses, Paris 2008.

Crossrefs

Cf. A001222, A045920, A052215, A093548.

Equivalent sequences for longer runs: A113752 (3), A356893 (4).

Programs

Maple

f:= proc(n) uses priqueue; local t,x,p,i;
    initialize(pq);
    insert([-3^n, 3$n], pq);
    do
      t:= extract(pq);
      x:= -t[1];
      if numtheory:-bigomega(x-1)=n then return x-1
      elif numtheory:-bigomega(x+1)=n then return x
      fi;
      p:= nextprime(t[-1]);
      for i from n+1 to 2 by -1 while t[i] = t[-1] do
        insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
      od;
    od
end proc:
seq(f(i),i=1..27); # Robert Israel, Sep 30 2024

Extensions

a(13)-a(15) from Martin Fuller, Jan 17 2006

a(16)-a(17) from Donovan Johnson, Apr 08 2008

a(18)-a(22) from Donovan Johnson, Jan 21 2009

a(23)-a(25) from Donovan Johnson, May 25 2013

a(26)-a(27) from Robert Israel, Sep 30 2024