A058033 -id:A058033 - cp's OEIS frontend

A084972 When A058033 first reaches n.

1, 2, 3, 4, 7, 12, 20, 35, 57, 102, 182, 323, 585, 1061, 1943, 3521, 6606

Offset: 1

Robert G. Wilson v, May 22 2003

nonn

Primorial[n_] := Product[Prime[i], {i, 1, n}]; f[n_] := Block[{k = 2, p = 1}, While[q = Floor[ Log[2, Primorial[k]]]; q - p != n, k++; p = q]; k - 1]; Table[ f[n], {n, 1, 17}]

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

Original entry on oeis.org

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

Lekraj Beedassy, May 22 2003

nonn

A measure of the growth rate of the primorials.

			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.

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

Cf. A000523, A002110, A058033.

Equals A045716(n) - 1.

a := n -> ilog2(mul(ithprime(i), i=1..n)):
seq(a(n), n=1..58); # Peter Luschny, Oct 18 2018

Table[Floor[Log[2, Product[Prime[i], {i, n}]]], {n, 60}]
Floor[Log2[#]]&/@FoldList[Times,Prime[Range[60]]] (* Harvey P. Dale, Aug 04 2021 *)

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

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

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).

A084321 Least number k such that between k! and (k+1)! there are n powers of 2 (each interval includes (k+1)! but not k!).

Original entry on oeis.org

1, 3, 5, 10, 19, 35, 64, 139, 256, 536, 1061, 2095, 4169, 8282, 16517, 32903, 65646, 131205, 262579, 525083, 1048893, 2098826, 4195521, 8390583, 16782032, 33560609, 67118347, 134229613, 268453180, 536890474, 1073764782, 2147523518

Offset: 1

Labos Elemer, Jun 19 2003

nonn

a(n) is near the (n-1)th power of 2, the difference is A085355.

			a(3)=5 since between 5!=120 and 6!=720 is the first time 3 powers of 2 arise, namely, 128, 256 and 512.

Kevin Ryde, Table of n, a(n) for n = 1..750
Kevin Ryde, C Code

Cf. A067850, A058033, A000142, A000079, A084320, A084420, A085355.

C
```
/* See links */
```

LogBase2Stirling[n_] := N[ Log[2, 2 Pi n]/2 + n*Log[2, n/E] + Log[2, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5)], 64]; k = 1; Do[ While[ Floor[ LogBase2Stirling[k + 1]] - Floor[ LogBase2Stirling[k]] < n, k++ ]; Print[k], {n, 1, 33}]

a(n) = minimum x for which floor(log_2((x+1)!)) - floor(log_2(x!)) = n.

a(n) = minimum x for which A084320(x) = n.

Edited and extended by Robert G. Wilson v, Jun 24 2003
Definition clarified by Jianing Song, Aug 08 2022
a(26) corrected by Kevin Ryde, Apr 25 2024

A084320 Number of powers of two between 2 consecutive factorials (2! including).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 5, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 7, 6, 6, 7, 6, 6, 7, 6, 7, 6, 7, 6, 6, 7, 7, 6, 7, 6, 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 6

Offset: 1

Labos Elemer, Jun 19 2003

nonn

			n=7: a(7)=3 because between 5040 and 40320 three powers of 2 occur: 8192, 16384 and 32768.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A067850, A058033, A000142, A000079.

Table[Floor[Log[2, (w+1)! ]//N]-Floor[Log[2, w! ]//N], {w, 1, 128}]

a(n)=if(n<6,(n+1)\2,log((n+1)!)\log(2)-log(n!)\log(2)) \\ Charles R Greathouse IV, Dec 26 2013

a(n) = A067850(n+1) - A067850(n).

a(n) = A000523(n) + O(1). - Charles R Greathouse IV, Dec 26 2013

cp's OEIS Frontend

A084972 When A058033 first reaches n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A084321 Least number k such that between k! and (k+1)! there are n powers of 2 (each interval includes (k+1)! but not k!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Mathematica

Formula

Extensions

A084320 Number of powers of two between 2 consecutive factorials (2! including).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula