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

3,1

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)

Links

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

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000040, A001358, A037143.

Sequence in context: A184160 A064553 A126012 * A096908 A162684 A265533

Adjacent sequences: A283264 A283265 A283266 * A283268 A283269 A283270

Keyword

nonn

Author

Vladimir Shevelev, Mar 04 2017

Extensions

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

Status

approved