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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

%I

%S 9,33,91,141,115,213,1382,1639,1133,2558,2973,1329,15685,16143,9974,

%T 35678,34063,43333,19613,107381,162145,44294,404599,461722,838259,

%U 155923,535403,492117,396737,2181739,370262,1468279,6034249,3933601,1671783,25180174,1357203

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

%C 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.

%C 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.

%C a(91) > 1.8*10^12. - _Giovanni Resta_, Aug 15 2013

%H Giovanni Resta, <a href="/A228170/b228170.txt">Table of n, a(n) for n = 1..90</a>

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

%e a(1) = 9 because between 9 and 10 there are no primes;

%e a(2) = 33 because between 33 and 35 (the second semiprime past 33) there are no primes;

%e a(3) = 91 because between 91 and 95 (the third semiprime past 91 with 93 & 94 also semiprimes) there are no primes;

%e a(4) = 141 because between 141 and 146 (the fourth semiprime past 141 with 142, 143 & 145 also being semiprimes) there are no primes;

%e 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.

%e 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;

%t 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

%Y Cf. A001358, A133478, A112888, A228171.

%K nonn

%O 1,1

%A _Jack Brennen_, _Jonathan Vos Post_, _Zak Seidov_, and _Robert G. Wilson v_, Aug 14 2013

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Last modified January 26 17:21 EST 2020. Contains 331280 sequences. (Running on oeis4.)