A283267 Smallest b-a such that a < prime(n) < b, where a,b are semiprimes.

2, 3, 4, 4, 6, 6, 3, 7, 7, 3, 7, 7, 3, 4, 4, 4, 4, 5, 5, 5, 3, 4, 11, 11, 11, 5, 5, 4, 6, 4, 7, 7, 9, 9, 3, 5, 3, 8, 5, 5, 7, 7, 7, 7, 4, 5, 9, 9, 9, 10, 10, 4, 5, 3, 7, 7, 4, 9, 9, 4, 4, 5, 5, 5, 5, 4, 9, 9, 9, 3, 6, 6, 4, 4, 5, 3, 5, 4, 5, 5, 10, 10, 8, 8, 4

Offset: 3

Author

Vladimir Shevelev, Mar 04 2017

Keywords

• nonn

Comments

This is the first sequence from the series of sequences: "Smallest b-a such that a < prime(n)^k < b, where a,b are semiprimes", k = 1, 2, 3, ... .

This series of sequences was inspired by Zak Seidov's message to Seqfans (Mar 02 2017) where he suggested listing the triples of primes squared with neighbor semiprimes.

There are no semiprimes below prime(2) = 3 but there are below prime(3) = 5 so the offset is 3. - David A. Corneth, Mar 04 2017

From Michael De Vlieger, Mar 04 2017: (Start)

Largest term in range a(3)..a(10^m): {7, 11, 24, 38, 54, 74, ...}.

Largest term in range a(3)..a(2^m), m>1: {3, 6, 7, 11, 11, 14, 19, 20, 24, 25, 38, 38, 38, 47, 47, 55, 70, 74, ...}.

Largest run in range a(3)..a(10^m): {2, 4, 6, 8, 10, 12, ...}.

Largest run in range a(3)..a(2^m), m>1: {1, 2, 2, 4, 4, 4, 4, 5, 6, 7, 7, 8, 8, 10, 10, 10, 12, 12, ...}. (End)

Examples

			For a(3), the largest semiprime below 5 is 4. The least semiprime above 5 is 6. Therefore, (a, b) = (4, 6) giving a(3) = 6 - 4 = 2. - _David A. Corneth_, Mar 04 2017
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A000040, A001358, A037143.

Programs

Mathematica

Table[Module[{p = Prime@ n, a, b}, a = p - 1; b = p + 1; While[PrimeOmega@ a != 2, a--]; While[PrimeOmega@ b != 2, b++]; b - a], {n, 3, 120}] (* Michael De Vlieger, Mar 04 2017 *)

PARI

issemi(n)=bigomega(n)==2
a(n,p=prime(n))=my(a=p,b=p); while(!issemi(a--), ); while(!issemi(b++), ); b-a \\ Charles R Greathouse IV, Mar 04 2017

Formula

a(n) = A102414(n) - A102415(n). - Michel Marcus, Mar 04 2017

Extensions

More terms from Peter J. C. Moses, Mar 04 2017