OFFSET
1,2
COMMENTS
A measure of the growth rate of the primorials.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = floor(log_2 n#) = m such that 2^m <= p(n)# < 2^(m + 1), where p(n)# is the primorial of the n-th prime (A002110).
a(n) ~ k*n log n, where k = 1/log(2) = A007525. - Charles R Greathouse IV, Sep 08 2025
EXAMPLE
The product of the first four primes is 2 * 3 * 5 * 7 = 210. In binary, 210 is 11010010, an 8-bit number, and we see that 2^7 < 210 < 2^8. And indeed log_2 210 = 7.7142455... and thus a(4) = 7.
a(5) = floor(log_2 2310) = floor(11.1736771363...) = 11.
MAPLE
a := n -> ilog2(mul(ithprime(i), i=1..n)):
seq(a(n), n=1..58); # Peter Luschny, Oct 18 2018
MATHEMATICA
Table[Floor[Log[2, Product[Prime[i], {i, n}]]], {n, 60}]
Floor[Log2[#]]&/@FoldList[Times, Prime[Range[60]]] (* Harvey P. Dale, Aug 04 2021 *)
PROG
(PARI) a(n) = logint(prod(k=1, n, prime(k)), 2); \\ Michel Marcus, Jan 06 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Lekraj Beedassy, May 22 2003
EXTENSIONS
Edited, corrected and extended by Robert G. Wilson v, May 22 2003
Name simplified by Alonso del Arte, Oct 14 2018 (old name is now first formula).
STATUS
approved