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A121406 a(1) = a(2) = 0; a(3) = 2; for n >= 4, a(n) = (prime(n-1)-2)*a(n-1), where prime(n) is the n-th prime.

%I A121406 #21 Feb 26 2024 01:55:01
%S A121406 0,0,2,6,30,270,2970,44550,757350,15904350,429417450,12453106050,
%T A121406 435858711750,16998489758250,696938080088250,31362213603971250,
%U A121406 1599472893802533750,91169954946744423750,5379027341857921001250
%N A121406 a(1) = a(2) = 0; a(3) = 2; for n >= 4, a(n) = (prime(n-1)-2)*a(n-1), where prime(n) is the n-th prime.
%C A121406 Also number of distinct twin prime eliminations which can be attributed to a particular lowest prime factor prime(n) over primorial intervals of prime(n)#. That is, it is the number of composite numbers having prime(n) for their lowest prime factor within any interval of width prime(n)# starting after prime(n) which are adjacent to the center post of a twin prime candidate for which that twin prime candidate is not also eliminated by a prime factor less than prime(n). Or put simply, it is the number of twin prime eliminations by prime(n) within intervals of its primorial that are left after subtracting out the double eliminations that can be attributed to previous prime factors.
%C A121406 Sum_{ n >= 1 } a(n)/A002110(n) converges to 1/6. That is, (2 / 30) + (6 / 210) + (30 / 2310) + (270 / 30030) + (2970 / 510510) + ... = (1 / 6).
%H A121406 Dennis Martin, <a href="/A121406/b121406.txt">Table of n, a(n) for n = 1..100</a>
%H A121406 Dennis R. Martin, <a href="http://www.primenace.com/papers/math/TwinPrimeEliminationSeries.htm">On the Infinite Series Characterizing the Elimination of Twin Prime Candidates</a> [Broken link]
%H A121406 Dennis R. Martin, <a href="/A121406/a121406.pdf">On the Infinite Series Characterizing the Elimination of Twin Prime Candidates</a> [Cached copy, with permission of the author]
%F A121406 a(1) = a(2) = 0; a(3) = 2; for n >= 4, a(n) = (prime(n-1)-2)*a(n-1), where prime(n) is the n-th prime.
%e A121406 The prime factors prime(1) = 2 and prime(2) = 3 cannot eliminate any twin prime candidates, therefore a(1) = a(2) = 0. The prime factor prime(3) = 5 will eliminate a(3) = 2 twin prime candidates in every interval of prime(3)# = 30 starting after prime(3). For example, the composites 25 and 35 eliminate the twin prime candidate pairs centered at 24 and 36, respectively, while the composites 55 and 65 eliminate the twin prime candidates centered at 54 and 66.
%e A121406 For the prime factor prime(4) = 7, there will be 8 composites having prime(4) for their lowest prime factor within every interval of prime(4)# = 210 starting after 7. For instance, the composites {49, 77, 91, 119, 133, 161, 203, 217} are adjacent to and eliminate the twin prime candidates centered at {48, 78, 90, 120, 132, 162, 204, 216}. However, 2 of those 8 are already eliminated by prime(3), those being the candidates centered at 204 and 216, since 205 and 215 obviously are composites having 5 for their lowest prime factor. And in the next interval of prime(4)# = 210 the pattern repeats. The composites {259, 287, 301, 329, 343, 371, 413, 427} all have 7 for their lowest prime factor and they eliminate the twin prime candidate pairs centered at {258, 288, 300, 330, 342, 372, 414, 426}. But the ones centered at 414 and 426 are also adjacent to 415 and 425, which have 5 for their lowest prime factor and thus can be considered to have already been eliminated. a(4) = 8 - 2 = 6.
%e A121406 For prime(5) = 11, there are 48 composites that have 11 for their lowest prime factor over any interval of prime(5)# = 2310 starting after 11. Those 48 composites are all adjacent to a twin prime candidate center post, but 12 of those candidates are eliminated by prime(3) (the ones corresponding to the centers 144, 186, 474, 516, 804, 1134, 1176, 1506, 1794, 1836, 2124 and 2166) and 6 are eliminated by prime(4) (those corresponding to the candidate centered at 120, 342, 582, 1728, 1968 and 2190). That is a total of 18 out of those 48 in every interval of 2310 that are eliminated by a prime factor less than prime(5), therefore a(5) = 48 - 18 = 30.
%e A121406 But then 30 = 6(7-2) and 6 = 2(5-2). By continuing to count the twin prime eliminations in this manner, it can be deduced that each subsequent term is found by multiplying the previous term by the previous prime minus 2.
%t A121406 nxt[{n_,a_}]:={n+1,a(Prime[n]-2)}; Join[{0,0},Transpose[NestList[nxt,{3,2},20]][[2]]] (* _Harvey P. Dale_, Sep 23 2015 *)
%Y A121406 Cf. A002110, A005867, A111490, A121407.
%K A121406 easy,frac,nonn
%O A121406 1,3
%A A121406 Dennis R. Martin (dennis.martin(AT)dptechnology.com), Jul 28 2006, Dec 05 2006