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%I #16 Dec 25 2020 20:08:08
%S 27,63,117,147,171,261,273,333,343,387,399,477,507,549,609,637,639,
%T 711,741,777,801,903,909,931,963,1017,1083,1113,1131,1179,1183,1251,
%U 1281,1359,1421,1443,1467,1491,1557,1629,1653,1659,1677,1729,1737,1791,1813,1869
%N Products of three prime numbers of even index.
%C All terms are odd.
%C Also Heinz numbers of integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.
%e The sequence of terms together with their prime indices begins:
%e 27: {2,2,2} 637: {4,4,6} 1183: {4,6,6}
%e 63: {2,2,4} 639: {2,2,20} 1251: {2,2,34}
%e 117: {2,2,6} 711: {2,2,22} 1281: {2,4,18}
%e 147: {2,4,4} 741: {2,6,8} 1359: {2,2,36}
%e 171: {2,2,8} 777: {2,4,12} 1421: {4,4,10}
%e 261: {2,2,10} 801: {2,2,24} 1443: {2,6,12}
%e 273: {2,4,6} 903: {2,4,14} 1467: {2,2,38}
%e 333: {2,2,12} 909: {2,2,26} 1491: {2,4,20}
%e 343: {4,4,4} 931: {4,4,8} 1557: {2,2,40}
%e 387: {2,2,14} 963: {2,2,28} 1629: {2,2,42}
%e 399: {2,4,8} 1017: {2,2,30} 1653: {2,8,10}
%e 477: {2,2,16} 1083: {2,8,8} 1659: {2,4,22}
%e 507: {2,6,6} 1113: {2,4,16} 1677: {2,6,14}
%e 549: {2,2,18} 1131: {2,6,10} 1729: {4,6,8}
%e 609: {2,4,10} 1179: {2,2,32} 1737: {2,2,44}
%t Select[Range[1000],PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]
%o (PARI) isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ _Michel Marcus_, Nov 10 2020
%Y A014612 allows all prime indices (not just even) (strict: A007304).
%Y A066207 allows products of any length (strict: A258117).
%Y A338471 is the version for odds instead of evens (strict: A307534).
%Y A338557 is the strict case.
%Y A014311 is a ranking of ordered triples (strict: A337453).
%Y A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
%Y A005117 lists squarefree numbers, with even case A039956.
%Y A008284 counts partitions by sum and length (strict: A008289).
%Y A023023 counts 3-part relatively prime partitions (strict: A101271).
%Y A046316 lists products of exactly three odd primes (strict: A046389).
%Y A066208 lists numbers with all odd prime indices (strict: A258116).
%Y A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
%Y A307719 counts 3-part pairwise coprime partitions (strict: A220377).
%Y A285508 lists Heinz numbers of non-strict triples.
%Y Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337599, A337600.
%Y Subsequence of A332820.
%K nonn
%O 1,1
%A _Gus Wiseman_, Nov 08 2020
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