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A226526 Slowest-growing sequence of semiprimes where 1/(sp+1) sums to 1 without actually reaching it.

%I A226526 #12 Sep 29 2013 20:18:39
%S A226526 4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,69,1497,259465,
%T A226526 4852747709,3429487924785490781,305153651313989042415043589313598477,
%U A226526 21932475414742921908206321699222250910796483151080020353252738457741771
%N A226526 Slowest-growing sequence of semiprimes where 1/(sp+1) sums to 1 without actually reaching it.
%C A226526 The semiprime analogous to A181503.
%C A226526 Because the semiprimes are sparser than the primes in the beginning, the sequence contains more of the lesser semiprimes than the analogous sequence of primes. In fact, one has to get to the seventeenth semiprime before it, 49,is not present, whereas in A181503, one only has to get to the sixth prime before it, 13, is not present.
%C A226526 If you change 1/(a(n)+1) to simply 1/a(n) the sequence becomes: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 355, 16627, 76723511, 17218740226618333, 374886275842473712491638217368219, 9036922116709843444667289331349853231276337589593114741410804131,....
%e A226526 1/(4+1) + 1/(6+1) + 1/(9+1) + … 1/(46+1) + 1/(69+1) is still less than 1. Instead of 1/69, if one were to use any semiprime between 46 and 69, {} the sum would then exceed 1.
%t A226526 semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2 (* For those who have Mmca v or later, you could use PrimeOmega@ n == 2 *) 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]]; a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextSemiPrime[ Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Do[ Print[{n, a[n] // Timing}], {n, 25}]
%Y A226526 Cf. A181503, A226527.
%K A226526 nonn,hard
%O A226526 1,1
%A A226526 _Aaron Meyerowitz_, _Jonathan Vos Post_, and _Robert G. Wilson v_, Jun 09 2013