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A242068 First of two consecutive sphenic numbers with no semiprime between them.

%I A242068 #17 Jan 26 2024 00:45:44
%S A242068 102,170,230,238,255,282,285,366,399,429,430,434,438,598,602,606,609,
%T A242068 615,638,642,645,651,663,741,759,805,822,826,854,902,935,969,986,1001,
%U A242068 1022,1030,1065,1085,1086,1102,1105,1130,1178,1182,1221,1245,1265,1295,1309,1310,1334,1358,1374,1406,1419,1426,1434
%N A242068 First of two consecutive sphenic numbers with no semiprime between them.
%C A242068 Sphenic numbers are products of three distinct primes.  Semiprimes are products of two primes, not necessarily distinct.
%C A242068 Contains A215217.
%H A242068 Robert Israel, <a href="/A242068/b242068.txt">Table of n, a(n) for n = 1..8683</a>
%e A242068 102=2*3*17 and 105=3*5*7 are sphenic numbers, i.e., products of three distinct primes, and neither 103 (a prime) nor 104=2^3*13 is a semiprime, so 102 is in the sequence.
%p A242068 N:= 10000: # to get all terms where the next sphenic number <= N
%p A242068 Sphenics:= select(t -> (map(s->s[2],ifactors(t)[2])=[1,1,1]), {$1..N}):
%p A242068 Primes:= select(isprime,{2,seq(2*i+1,i=1..floor(N/2))}):
%p A242068 Semiprimes:= {seq(seq(p*q,q=select(`<=`,Primes,N/p)),p=Primes)}:
%p A242068 map(proc(i) if nops(Semiprimes intersect {$Sphenics[i]..Sphenics[i+1]}) = 0 then Sphenics[i] else NULL fi end proc, [$1..nops(Sphenics)-1]);
%t A242068 sw = Switch[FactorInteger[#][[All, 2]], {1, 1}, {#, 2}, {1, 1, 1}, {#, 3}, _, Nothing]& /@ Range[10^4];
%t A242068 sp = SequencePosition[sw, {{_, 3}, {_, 3}}][[All, 1]];
%t A242068 sw[[sp]][[All, 1]] (* _Jean-François Alcover_, Sep 26 2020 *)
%Y A242068 Cf. A001358, A007304, A215217.
%K A242068 nonn
%O A242068 1,1
%A A242068 _Robert Israel_, Aug 13 2014