A131939 Least k such that the difference between consecutive 3-almost primes A014612(k) equals n, or 0 if no such k exists.

5, 3, 13, 1, 10, 2, 4, 31, 32, 36, 12, 7, 136, 19, 302, 486, 1094, 73, 1366, 6763, 1092, 2006, 8924, 4785, 18345, 18487, 42798, 16571, 11095, 57831, 60912, 4528, 24846, 41304, 232350, 233678, 123279, 1779265, 740729, 177385, 1015228, 1772286

Offset: 1

Author

Jonathan Vos Post, Oct 05 2007

Keywords

• nonn

Comments

Index of smallest 3-almost prime gap equal to n. This is to 3-almost primes A014612 as A123375 is to semiprimes A001358. a(25) = 0 or a(25) > 10000. Conjecture: a(n) > 0 for all n.

Examples

			a(1) = 5 because A014612(6)-A014612(5) = 28-27 = 1.
a(2) = 3 because A014612(4)-A014612(3) = 20-18 = 2.
a(3) = 13 because 66-63 = 3.
a(4) = 1 because 12-8 = 4.
a(5) = 10 because 50-45 = 5.
a(6) = 2 because 18-12 = 6.
a(7) = 4 because 27-20 = 7.
a(8) = 31 because 138-130 = 8.
a(9) = 32 because 147-138 = 9
a(10) = 36 because 164-154 = 10.
		

Crossrefs

Cf. A014612, A114403, A123375.

Programs

Mathematica

p3=Select[Range[2*10^6],PrimeOmega[#]==3&];s={};Do[d=0;Until[p3[[d+1]]-p3[[d]]==n,d++];AppendTo[s,d],{n,37}];s (* James C. McMahon, Mar 02 2025 *)

Formula

a(n) = MIN{k such that A114403(k) = n, or 0 if no such k exists}.

a(n) = MIN{k such that A014612(k+1) - A014612(k) = n, or 0 if no such k exists}.

Extensions

More terms from R. J. Mathar, Oct 07 2007