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).
There is a more specific property we could use in our definition: a(n) is the numerator of the proportion of numbers whose third smallest prime divisor both exists and is prime(n) in every interval of A002110(n) positive integers, where A002110(n) is the product of the first n primes. The number in each such interval is A096294(n-1, 2), n > 2. Compare with column 3 of A281890. - Peter Munn, Apr 03 2026
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
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.
Sum_{n>=1} a(n)/A378721(n) = 1.
For n > 2, a(n) = count(n)/gcd(count(n), A002110(n)), where count(n) = A096294(n-1, 2). - Peter Munn, Apr 03 2026
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
AUTHOR
Robert G. Wilson v and Amiram Eldar, Dec 05 2024
STATUS
approved