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

102, 170, 230, 238, 255, 282, 285, 366, 399, 429, 430, 434, 438, 598, 602, 606, 609, 615, 638, 642, 645, 651, 663, 741, 759, 805, 822, 826, 854, 902, 935, 969, 986, 1001, 1022, 1030, 1065, 1085, 1086, 1102, 1105, 1130, 1178, 1182, 1221, 1245, 1265, 1295, 1309, 1310, 1334, 1358, 1374, 1406, 1419, 1426, 1434

Offset: 1

Author

Robert Israel, Aug 13 2014

Keywords

• nonn

Comments

Sphenic numbers are products of three distinct primes. Semiprimes are products of two primes, not necessarily distinct.

Contains A215217.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..8683

Crossrefs

Cf. A001358, A007304, A215217.

Programs

Maple

N:= 10000: # to get all terms where the next sphenic number <= N
Sphenics:= select(t -> (map(s->s[2],ifactors(t)[2])=[1,1,1]), {$1..N}):
Primes:= select(isprime,{2,seq(2*i+1,i=1..floor(N/2))}):
Semiprimes:= {seq(seq(p*q,q=select(`<=`,Primes,N/p)),p=Primes)}:
map(proc(i) if nops(Semiprimes intersect {$Sphenics[i]..Sphenics[i+1]}) = 0 then Sphenics[i] else NULL fi end proc, [$1..nops(Sphenics)-1]);

Mathematica

sw = Switch[FactorInteger[#][[All, 2]], {1, 1}, {#, 2}, {1, 1, 1}, {#, 3}, _, Nothing]& /@ Range[10^4];
sp = SequencePosition[sw, {{, 3}, {, 3}}][[All, 1]];
sw[[sp]][[All, 1]] (* Jean-François Alcover, Sep 26 2020 *)