cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Lekraj Beedassy, May 22 2003

Keywords

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

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).

A058033 Number of powers of 2 between (but not including) two consecutive primorials.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 5, 4, 5, 5, 5, 6, 5, 6, 5, 6, 6, 6, 6, 7, 6, 6, 7, 6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 8, 7, 8, 7, 8, 7, 8, 8, 7, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 9, 8, 8, 8, 8, 8, 9, 8, 8, 9, 8, 9, 8, 8, 9, 8, 9, 9, 8, 9, 8, 9, 9, 8, 9, 9, 9, 8, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9
Offset: 1

Views

Author

Labos Elemer, Nov 22 2000

Keywords

Comments

Does not increase monotonically.

Examples

			Between [2 and 6] is 4;
between [6 and 30] are 8, 16;
between [30 and 210] are 32, 64, 128;
between [210 and 2310] are 256, 512, 1024, 2048;
between [2310 and 30030] are 4096, 8192, 16384;
between [30030, 510510] are 32768, 65536, 131072, 262144.
So a(1), ..., a(6) = 1, 2, 3, 4, 3, 4, ...
From _Michael De Vlieger_, Jul 15 2017: (Start)
First and last positions of values seen in the first 10^5 terms:
Value  First   Last
   1      1       1
   2      2       2
   3      3       5
   4      4       8
   5      7      15
   6     12      27
   7     20      51
   8     35      90
   9     57     161
  10    102     294
  11    182     542
  12    323     995
  13    585    1856
  14   1061    3505
  15   1943    6485
  16   3521   12203
  17   6606   22949
  18  12297   43200
  19  23051   81759
  20  43578  (99999)
  21  82296  (99997)
(End)

Links

Crossrefs

Cf. A002110, A045716, A054850, A084972.

Programs

  • Mathematica
    a = Table[ Floor[ Log[2, Product[ Prime[i], {i, 1, n}]]], {n, 1, 110}]; Drop[a, 1] - Drop[a, -1]
(* Second program: *)
Differences@ Floor@ Log2@ FoldList[Times, Prime@ Range@ 105] (* Michael De Vlieger, Jul 15 2017 *)

Formula

a(n)= Cardinality{b |A002110(n)<= b < A002110(n+1), b=2^x for some x} a(n)=A045716(n+1)-A045716(n)
First differences of A054850.

Extensions

Edited by Robert G. Wilson v, May 22 2003

A061717 Binary order of n^n.

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 20, 24, 29, 34, 39, 44, 49, 54, 59, 64, 70, 76, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 148, 154, 160, 167, 173, 180, 187, 193, 200, 207, 213, 220, 227, 234, 241, 248, 255, 262, 269, 276, 283, 290, 297, 304, 311, 318, 326, 333
Offset: 1

Views

Author

Labos Elemer, Jun 20 2001

Keywords

Links

Crossrefs

Cf. A000312, A029837, A003070, A045716.

Programs

  • PARI
    a(n) = { if(n<=1, 0, logint(n^n-1, 2) + 1) } \\ Harry J. Smith, Jul 26 2009

Formula

a(n) = ceiling(log_2(n^n)) = A029837(A000312(n)).

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Jul 26 2009

A046971 Maximal value of number of unitary divisors (see A034444) for integers in binary order range of n.

Original entry on oeis.org

2, 2, 4, 4, 8, 8, 8, 16, 16, 16, 16, 32, 32, 32, 64, 64, 64, 64, 128, 128, 128, 128, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512, 1024, 1024, 1024, 1024, 1024, 2048, 2048, 2048, 2048, 2048, 4096, 4096, 4096, 4096, 4096, 4096, 8192, 8192, 8192, 8192
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Q_k the k-th primorial number (A002110) is the smallest positive integer whose number of unitary divisors (A034444) is 2^k. Thus the binary orders (A029837) of primorial numbers (A045716) determine the maximal values of unitary divisors in binary order ranges.

Examples

			128 occurs in the sequence from index 19 and 23 inclusively (5 times) because the 7th primorial number Q_7=510510 has binary order 19 and Q_8, the next one is 9699690 has binary order 24 (it is between 2^23 and 2^24).

Links

Crossrefs

Cf. A002110, A029837, A034444, A045716.

Programs

  • PARI
    v=List([2]);n=0;P=p=2;forprime(q=3,97, n++;for(i=1, log(P*q)\log(2)-log(P)\log(2), listput(v,1<Charles R Greathouse IV, Apr 08 2012

Formula

2^k occurs in the sequence g(Q_k) - g(Q_k-1)+1 times, where g is the binary order and Q_k is the k-th primorial number.

A058032 Largest m such that 2^n / primorial(m) >= 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18
Offset: 0

Views

Author

Labos Elemer, Nov 22 2000

Keywords

Comments

Primorial order of powers of 2.

Examples

			For n = 1 and 2, when 2 and 4 are divided by 2 the quotient = 1 or 2, but when they are divided by 6 the quotient < 1, so the largest suitable primorial is the first; thus a(1) = a(2) = 1.
For n = 11, 2^11 = 2048. The largest primorial P, such that 2048/P > 1 is P = 210, the 4th = A002110(4). So a(11) = 4.

Links

Crossrefs

Cf. A045716 (binary order, A029837, of primorials, A002110).

Programs

  • Mathematica
    seq[lim_] := Module[{m = -1, p = 1, primo = 1, pow = 1, s = {}}, Do[While[pow < primo, AppendTo[s, m]; pow *= 2]; p = NextPrime[p]; primo *= p; m++, {lim}]; s]; seq[20] (* Amiram Eldar, Feb 08 2025 *)

Formula

a(n) = Max{m| sign(floor(2^n/A002110(m))) = 1}.

Extensions

a(0) inserted by Amiram Eldar, Feb 08 2025

A061716 Binary order of n-th prime.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10
Offset: 1

Views

Author

Labos Elemer, Jun 20 2001

Keywords

Comments

Apart from the first terms, the same as A035100. - R. J. Mathar, Oct 02 2008

Links

Crossrefs

Cf. A029837, A000040, A003070, A045716, A045716, A036378.

Programs

  • Mathematica
    Ceiling[Log2[Prime[Range[110]]]] (* Harvey P. Dale, Apr 12 2023 *)
  • PARI
    a(n) = { logint(prime(n)-1, 2) + 1 } \\ Harry J. Smith, Jul 26 2009

Formula

a(n) = ceiling(log_2(prime(n))) = A029837(A000040(n)).

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Jul 26 2009
Showing 1-6 of 6 results.

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