OFFSET
1,5
COMMENTS
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.
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).
REFERENCES
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..365
Jean-Marie de Koninck and Gérald Tenenbaum, Sur la loi de répartition du k-ième facteur premier d'un entier, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 133, No. 2 (2002), pp. 191-204.
Paul Erdős and Gérald Tenenbaum, Sur les densités de certaines suites d'entiers, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; alternative link.
FORMULA
EXAMPLE
The fractions begin with 0/1, 0/1, 1/30, 1/30, 4/165, 326/15015, 628/36465, 992/62985, 98304/7436429, 125568/11849255, ..., .
a(1) = a(2) = 0 since there are no numbers whose third prime divisor is 2 or 3.
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.
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.
MATHEMATICA
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]
PROG
(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); }
CROSSREFS
KEYWORD
nonn,easy,frac,new
AUTHOR
Robert G. Wilson v and Amiram Eldar, Dec 05 2024
STATUS
approved