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A378720 a(n) is the numerator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).

%I A378720 #18 Apr 14 2025 05:35:49
%S A378720 0,0,1,1,4,326,628,992,98304,125568,733440,281163264,386427322368,
%T A378720 3178249003008,12454223855616,6450728943845376,342348724735967232,
%U A378720 20218431581110665216,39814891891080560640,82739188294287768944640,15336676441718784000,61298453882755419734016000
%N A378720 a(n) is the numerator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).
%C A378720 The third smallest prime divisor of a number k is the third member in the ordered list of the distinct prime divisors of k. Only numbers in A000977 have a third smallest prime divisor.
%C A378720 The partial sums of the fractions first exceed 1/2 after summing 4467 terms. Therefore, the median value of the distribution of the third prime divisor is prime(4467) = 42719 = A284411(3).
%D A378720 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.
%H A378720 Amiram Eldar, <a href="/A378720/b378720.txt">Table of n, a(n) for n = 1..365</a>
%H A378720 Jean-Marie de Koninck and Gérald Tenenbaum, <a href="https://doi.org/10.1017/S0305004102005972">Sur la loi de répartition du k-ième facteur premier d'un entier</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 133, No. 2 (2002), pp. 191-204.
%H A378720 Paul Erdős and Gérald Tenenbaum, <a href="https://doi.org/10.1112/plms/s3-59.3.417">Sur les densités de certaines suites d'entiers</a>, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; <a href="https://users.renyi.hu/~p_erdos/1989-36.pdf">alternative link</a>.
%F A378720 a(n)/A378721(n) = (1/prime(n)#) * (Product_{k=1..n-1} (prime(k) - 1)) * Sum_{j=1..n-1, i=1..j-1} 1/((prime(i)-1)*(prime(j)-1)), where prime(n)# = A002110(n) is the n-th primorial number.
%F A378720 Sum_{n>=1} a(n)/A378721(n) = 1.
%F A378720 Sum_{n=1..m} a(n)/A378721(n) > 1/2 for m >= 4467 = primepi(A284411(3)).
%e A378720 The fractions begin with 0/1, 0/1, 1/30, 1/30, 4/165, 326/15015, 628/36465, 992/62985, 98304/7436429, 125568/11849255, ..., .
%e A378720 a(1) = a(2) = 0 since there are no numbers whose third prime divisor is 2 or 3.
%e A378720 a(3)/A378721(3) = 1/30 since the numbers whose third prime divisor is 5 are the numbers that are divisible by 2, 3 and 5, and their density if (1/2)*(1/3)*(1/5) = 1/30.
%e A378720 a(4)/A378721(4) = 1/30 since the numbers whose third prime divisor is 7 are the union of the numbers that are divisible by 2, 3 and 7 and not by 5 whose density is (1/2)*(1/3)*(1-1/5)*(1/7) = 2/105, the numbers that are divisible by 2, 5 and 7 and not by 3 whose density is (1/2)*(1-1/3)*(1/5)*(1/7) = 1/105, and the numbers that are divisible by 3, 5 and 7 and not by 2 whose density is (1-1/2)*(1/3)*(1/5)*(1/7) = 1/210, and 2/105 + 1/105 + 1/210 = 1/30.
%t A378720 a[n_] := Block[{p, q = Prime@ Range@ n}, p = Fold[Times, 1, q]; q = Most@ q; Plus @@ Times @@@ Subsets[q -1, {n -3}]/p]; a[1] = 0; Numerator@ Array[a, 22]
%o A378720 (PARI) a(n) = {my(v = primes(n), q = vecextract(apply(x -> x-1, v),"^-1"), p = vecprod(v), prd = vecprod(q)/p, sm = 0, sb); forsubset([#q, 2], s, sb = vecextract(q, s); sm += 1/vecprod(sb)); numerator(prd * sm);}
%Y A378720 Cf. A000040, A002110, A000977, A038110, A038111, A284411, A342479, A342480, A378721 (denominators).
%K A378720 nonn,easy,frac
%O A378720 1,5
%A A378720 _Robert G. Wilson v_ and _Amiram Eldar_, Dec 05 2024