A067021 Largest prime of which the square still does not exceed the product of first n primes, the n-th primorial.

2, 5, 13, 47, 173, 709, 3109, 14929, 80429, 447829, 2724079, 17442769, 114379879, 784149077, 5708691479, 43849291271, 342473913367, 2803269796331, 23620771158583, 201815957246317, 1793779464521953, 16342108667160251, 154171144824008969, 1518409682511777919, 15259828451149028543

Offset

2,1

Links

Table of n, a(n) for n=2..26.

Formula

a(n) = Max[p; p^2 < A002110(n)], where p is prime; prime(n+s)^2 = a(n)^2 < Product[prime(1), ..., prime(n)] < prime(n+s+1)^2.

Example

For n=2, 3, 4, 5, 7: {2^2, 6, 3^2}, {5^2, 30, 7^2}, {13^2, 210, 17^2}, {47^2, 2310, 53^2} {709^2, 510510, 719^2} or {4, 6, 9}, {25, 30, 49}, {169, 210, 289}, {2209, 2310, 2809}, {502681, 510510, 516961}.
Also, if n=2, then a(2) < prime(1)=2, if n=3, then a(3)=prime(3)=5 but for n > 3, a(n) > prime(n+1); e.g., a(6) = prime(40) = prime(6+34) = 173.

Mathematica

q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] rq[x_] := Floor[Sqrt[q[w]]//N] Table[Prime[PrimePi[a[w]]], {w, 2, 15}]

Prog

(PARI) primo(n) = prod(i=1, n, prime(i)); \\ A002110
a(n) = precprime(sqrtint(primo(n))); \\ Michel Marcus, Aug 29 2019

Crossrefs

Cf. A002110, A067022.

Sequence in context: A360513 A194635 A212821 * A269068 A098716 A082938

Adjacent sequences: A067018 A067019 A067020 * A067022 A067023 A067024

Keyword

nonn

Author

Labos Elemer, Dec 29 2001

Extensions

More terms from Michel Marcus, Aug 29 2019

Status

approved