A054850 Binary logarithm of n-th primorial, rounded down to an integer.

1, 2, 4, 7, 11, 14, 18, 23, 27, 32, 37, 42, 48, 53, 59, 64, 70, 76, 82, 88, 95, 101, 107, 114, 120, 127, 134, 140, 147, 154, 161, 168, 175, 182, 189, 197, 204, 211, 219, 226, 234, 241, 249, 256, 264, 272, 279, 287, 295, 303, 311, 318, 326, 334, 342, 350, 358, 367

Offset: 1

Author

Lekraj Beedassy, May 22 2003

Keywords

• nonn

Comments

A measure of the growth rate of the primorials.

Examples

			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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000523, A002110, A058033.

Equals A045716(n) - 1.

Programs

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 *)

PARI

a(n) = logint(prod(k=1, n, prime(k)), 2); \\ Michel Marcus, Jan 06 2020

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) = A000523(A002110(n)).

a(n) ~ n log n. - Charles R Greathouse IV, Aug 25 2024

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).