A228170 The least semiprime (A001358) such that between it and the next n semiprimes, but not the next n+1 semiprimes, there are no primes.

9, 33, 91, 141, 115, 213, 1382, 1639, 1133, 2558, 2973, 1329, 15685, 16143, 9974, 35678, 34063, 43333, 19613, 107381, 162145, 44294, 404599, 461722, 838259, 155923, 535403, 492117, 396737, 2181739, 370262, 1468279, 6034249, 3933601, 1671783, 25180174, 1357203

Offset: 1

Author

Jack Brennen, Jonathan Vos Post, Zak Seidov, and Robert G. Wilson v, Aug 14 2013

Keywords

• nonn

Comments

If prime_omega(n) as defined as A001222 and a set of values becomes a string, then the 'just' means that its string is not a substring of some larger string. See the example below.

Yet another way to think of this is that between any two consecutive primes there are 'just' n semiprimes with the first one being cited above.

a(91) > 1.8*10^12. - Giovanni Resta, Aug 15 2013

Examples

			a(1) = 9 because between 9 and 10 there are no primes;
a(2) = 33 because between 33 and 35 (the second semiprime past 33) there are no primes;
a(3) = 91 because between 91 and 95 (the third semiprime past 91 with 93 & 94 also semiprimes) there are no primes;
a(4) = 141 because between 141 and 146 (the fourth semiprime past 141 with 142, 143 & 145 also being semiprimes) there are no primes;
the reason a(4) is not 115 is because although there are no primes between 115 and 121, the string "2, 3, 3, 2, 2, 5, 2, 2" is a substring of the string generated by 115 through 123. See the next line.
a(5) = 115 because between 115 and 123 (the fifth semiprime past 115 with 118, 119, 121, and 122 also being semiprimes) there are no primes;
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..90

Crossrefs

Cf. A001358, A133478, A112888, A228171.

Programs

Mathematica

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; t = Table[0, {100}]; p=3; While[p < 3100000000, q = NextPrime[p]; a = Count[ PrimeOmega[ Range[p, q]], 2]; If[ t[[a]] == 0, t[[a]] = p; Print[{p, a}]]; p = q]; NextSemiPrime@# & /@ t

Formula

a(n) is the next semiprime after A228171(n+1).