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

1,2

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) = A000523(A002110(n)).

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

Cf. A000523, A002110, A058033.

Equals A045716(n) - 1.

Sequence in context: A138766 A192638 A211372 * A225154 A167805 A027427

Adjacent sequences: A054847 A054848 A054849 * A054851 A054852 A054853

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