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

0, 0, 1, 1, 4, 326, 628, 992, 98304, 125568, 733440, 281163264, 386427322368, 3178249003008, 12454223855616, 6450728943845376, 342348724735967232, 20218431581110665216, 39814891891080560640, 82739188294287768944640, 15336676441718784000, 61298453882755419734016000

Offset: 1

Author

Robert G. Wilson v and Amiram Eldar, Dec 05 2024

Keywords

• nonn
• easy
• frac

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).

Examples

			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.
		

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.

Crossrefs

Cf. A000040, A002110, A000977, A038110, A038111, A284411, A342479, A342480, A378721 (denominators).

Programs

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]

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);}

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.

Sum_{n=1..m} a(n)/A378721(n) > 1/2 for m >= 4467 = primepi(A284411(3)).