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Binary logarithm of n-th primorial, rounded down to an integer.
8

%I #25 Aug 25 2024 09:49:39

%S 1,2,4,7,11,14,18,23,27,32,37,42,48,53,59,64,70,76,82,88,95,101,107,

%T 114,120,127,134,140,147,154,161,168,175,182,189,197,204,211,219,226,

%U 234,241,249,256,264,272,279,287,295,303,311,318,326,334,342,350,358,367

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

%C A measure of the growth rate of the primorials.

%H Michael De Vlieger, <a href="/A054850/b054850.txt">Table of n, a(n) for n = 1..10000</a>

%F 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).

%F a(n) = A000523(A002110(n)).

%F a(n) ~ n log n. - _Charles R Greathouse IV_, Aug 25 2024

%e 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.

%e a(5) = floor(log_2 2310) = floor(11.1736771363...) = 11.

%p a := n -> ilog2(mul(ithprime(i), i=1..n)):

%p seq(a(n), n=1..58); # _Peter Luschny_, Oct 18 2018

%t Table[Floor[Log[2, Product[Prime[i], {i, n}]]], {n, 60}]

%t Floor[Log2[#]]&/@FoldList[Times,Prime[Range[60]]] (* _Harvey P. Dale_, Aug 04 2021 *)

%o (PARI) a(n) = logint(prod(k=1, n, prime(k)), 2); \\ _Michel Marcus_, Jan 06 2020

%Y Cf. A000523, A002110, A058033.

%Y Equals A045716(n) - 1.

%K nonn

%O 1,2

%A _Lekraj Beedassy_, May 22 2003

%E Edited, corrected and extended by _Robert G. Wilson v_, May 22 2003

%E Name simplified by _Alonso del Arte_, Oct 14 2018 (old name is now first formula).