

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.


2



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
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OFFSET

1,1


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


LINKS

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


FORMULA

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


EXAMPLE

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;


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


CROSSREFS

Cf. A001358, A133478, A112888, A228171.
Sequence in context: A005939 A020326 A201024 * A112888 A048479 A031880
Adjacent sequences: A228167 A228168 A228169 * A228171 A228172 A228173


KEYWORD

nonn


AUTHOR

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


STATUS

approved



