%I #8 Sep 29 2013 20:18:20
%S 8,12,18,20,27,28,30,42,44,45,50,52,63,66,68,70,75,76,78,92,98,99,102,
%T 105,110,114,116,117,124,125,130,138,147,148,153,154,164,165,170,171,
%U 172,174,175,182,186,188,190,195,207,212,222,230,231,236,238,242,244,245,246,255,258,261,266,268,273,275,279,282,284,285,286,290,292,310,316,318,322,325,332,333,338,343,345,354,356,357,363,366,369,370,374,385,387,388,399,402,404,406,410,412,418,423,425,426,428,429,430,434,435,436,8662,44335708,1251938572491943,1505273212784203338150808798466,680617602541158152398258079780439819108542271775727566330763
%N Slowest-growing sequence of 3-almost primes (trientprimes) where 1/(tp+1) sums to 1 without actually reaching it.
%C See comments in A226526.
%C Deviates from A014612 after the 110th term.
%t kAlmostPrimeQ[n_, k_: 2] := Plus @@ Last /@ FactorInteger@ n == k (* For those who have Mmca v or later, you could use PrimeOmega@ n == k *) NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextkAlmostPrime[ Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Do[ Print[{n, a[n] // Timing}], {n, 25}]
%Y Cf. A181503, A226526.
%K nonn,hard
%O 1,1
%A _Aaron Meyerowitz_, _Jonathan Vos Post_, and _Robert G. Wilson v_, Jun 09 2013