A054850 -id:A054850 - cp's OEIS frontend

A045716 a(n) is the binary order (A029837) of the n-th primorial number, A002110(n).

1, 3, 5, 8, 12, 15, 19, 24, 28, 33, 38, 43, 49, 54, 60, 65, 71, 77, 83, 89, 96, 102, 108, 115, 121, 128, 135, 141, 148, 155, 162, 169, 176, 183, 190, 198, 205, 212, 220, 227, 235, 242, 250, 257, 265, 273, 280, 288, 296, 304, 312, 319, 327, 335, 343, 351, 359

Offset: 1

Labos Elemer

nonn

These orders determine the maximal numbers of unitary divisors valid for integers in given binary order ranges (see A046971).

			The sixth primorial number is 2*3*5*7*11*13 = 30030, which is in the interval [16385, 32768] = [2^14 + 1, 2^15], so its binary order is a(6)=15. [corrected by _Jon E. Schoenfield_, May 13 2018]

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

Cf. A029837, A002110, A046971, A054850.

Table[Ceiling@ Log2[Times @@ Prime@ Range@ n], {n, 57}] (* Michael De Vlieger, Feb 05 2017 *)

a(n)=logint(prod(i=1,n,prime(i))-1,2)+1 \\ Charles R Greathouse IV, Feb 06 2017

a(n) = A054850(n)+1, n >= 2.

a(n) = A070939(A002110(n)) for n>1. - Jachym Barvinek, Mar 22 2018

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

Labos Elemer, Nov 22 2000

nonn

Does not increase monotonically.

			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)

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

Cf. A002110, A045716, A054850, A084972.

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 *)

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.

Edited by Robert G. Wilson v, May 22 2003

A082811 Smallest multiple of n beginning with 2.

Original entry on oeis.org

2, 2, 21, 20, 20, 24, 21, 24, 27, 20, 22, 24, 26, 28, 210, 208, 204, 216, 209, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 210, 217, 224, 231, 204, 210, 216, 222, 228, 234, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 200, 204, 208, 212, 216, 220

Offset: 1

Amarnath Murthy, Apr 19 2003

a(n) is in {n, 2n, 3n, 4n, 5n, 6n, 7n, 11n, 12n, 13n, 14n}. [Charles R Greathouse IV, Mar 06 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A082784, A054850, A082786, A082787, A082788, A082789, A082791.

Cf. A000030.

a082811 n = until ((== 2) . a000030) (+ n) n
-- Reinhard Zumkeller, Mar 27 2012

f[n_] := Block[{m = n}, While[ First@ IntegerDigits@ m != 2, m += n]; m]; Array[f, 55] (* Robert G. Wilson v *)

a(n)=forstep(k=n, 14*n, n, if(Vec(Str(k))[1]=="2", return(k))) \\ Charles R Greathouse IV, Mar 06 2011

def A082811(n): return next(k for k in PositiveIntegers() if (n*k).digits()[-1] == 2) # D. S. McNeil, Mar 07 2011

A082791 Smallest k such that k*n begins with 2: a(n) = A082811(n)/n.

Original entry on oeis.org

2, 1, 7, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 14, 13, 12, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Amarnath Murthy, Apr 19 2003

a(n) is in {1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14}. [Charles R Greathouse IV, Mar 06 2011]

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A082784, A054850, A082786, A082787, A082788, A082789, A082811.

a(n)=forstep(k=n, 14*n, n, if(Vec(Str(k))[1]=="2", return(k/n))) \\ Charles R Greathouse IV, Mar 06 2011

More terms from Zak Seidov, Mar 06 2011

A287010 Triangle T(n,m): floor(log(A002110(n))/log(prime(m))).

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 7, 4, 3, 2, 11, 7, 4, 3, 3, 14, 9, 6, 5, 4, 4, 18, 11, 8, 6, 5, 5, 4, 23, 14, 9, 8, 6, 6, 5, 5, 27, 17, 11, 9, 8, 7, 6, 6, 6, 32, 20, 14, 11, 9, 8, 7, 7, 7, 6, 37, 23, 16, 13, 10, 10, 9, 8, 8, 7, 7, 42, 26, 18, 15, 12, 11, 10, 10, 9, 8, 8, 8

Offset: 1

Michael De Vlieger, Aug 31 2017

Row n lists the largest power e of the prime divisors p_m of primorial p_n# such that p_m^e <= p_n#.

			For n = 3, A002110(n) = 30 = 2 * 3 * 5; floor(log_2(30)) = 4, floor(log_3(30)) = 3, floor(log_5(30)) = 2, thus row 3 = {4, 3, 2}.
Triangle begins:
   1:   1
   2:   2   1
   3:   4   3   2
   4:   7   4   3   2
   5:  11   7   4   3   3
   6:  14   9   6   5   4   4
   7:  18  11   8   6   5   5   4
   8:  23  14   9   8   6   6   5   5
   9:  27  17  11   9   8   7   6   6   6
  10:  32  20  14  11   9   8   7   7   7   6
  11:  37  23  16  13  10  10   9   8   8   7   7
  12:  42  26  18  15  12  11  10  10   9   8   8   8
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows 1 <= n <= 150).

Cf. A002110, A054850.

Table[With[{P = Product[Prime@ i, {i, n}]}, Floor@ Log[Prime@ #, P] & /@ Range@ n], {n, 20}] // Flatten

A307133 T(n,m) = number of k <= A002110(n) such that A001221(k) = m, where k is a term in A025487.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 9, 4, 1, 1, 11, 21, 15, 5, 1, 1, 14, 38, 36, 18, 6, 1, 1, 18, 64, 79, 53, 23, 7, 1, 1, 23, 97, 148, 122, 63, 26, 7, 1, 1, 27, 140, 258, 251, 157, 76, 30, 7, 1, 1, 32, 196, 425, 480, 349, 195, 89, 33, 8, 1, 1, 37, 261, 655, 853

Offset: 0

Michael De Vlieger, Mar 26 2019

Terms m in A025487 are products of p_i# in A002110.

The primorial A002110(n) is the smallest number k that is the product of the n smallest primes (i.e., A001221(k) = n) and is a subset of A025487.

			Row 3 = {1,4,3,1}. The terms k in A025487 such that k <= A002110(3) are {1, 2, 4, 6, 8, 12, 16, 24, 30}. Of these, 1 has 0 distinct prime divisors, 4 {2,4,8,16} have 1 distinct prime divisor, 3 {6,12,24} have 2 distinct prime divisors, and 1 {30} has 3 distinct prime divisors.
Triangle begins:
   0: 1
   1: 1   1
   2: 1   2    1
   3: 1   4    3    1
   4: 1   7    9    4     1
   5: 1  11   21   15     5     1
   6: 1  14   38   36    18     6    1
   7: 1  18   64   79    53    23    7    1
   8: 1  23   97  148   122    63   26    7    1
   9: 1  27  140  258   251   157   76   30    7    1
  10: 1  32  196  425   480   349  195   89   33    8   1
  11: 1  37  261  655   853   700  443  228  102   37   9   1
  12: 1  42  340  975  1438  1323  928  533  268  119  41  11   1
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..560 (rows 0 <= n <= 1000).

Cf. A000012, A001221, A002110, A025487, A054850, A098719.

Block[{nn = 12, f, w}, f[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1], g}, g[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; Map[Block[{w = #, k = 1}, Sort@ Prepend[If[Length@ # == 0, #, #[[1]]], Product[Prime@ i, {i, Length@ w}]] &@ Reap[Do[If[# < lim, Sow[#]; k = 1, If[k >= Length@ w, Break[], k++]] &@ g@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1]]], {i, Infinity}]][[-1]]] &, ww]]; s = MapAt[Flatten, f@ nn, 1]; Array[Function[P, TakeWhile[Map[Count[#, _?(# <= P &)] &, s, {1}], # > 0 &]]@ Product[Prime@ i, {i, #}] &, nn + 1, 0]] // Flatten

T(n,0) = T(n,n) = A000012(n).
T(n,1) = A054850(n).
A098719(n) = sum of row n.

A319880 Difference between 2^n and the product of primes less than or equal to n.

Original entry on oeis.org

0, 1, 2, 2, 10, 2, 34, -82, 46, 302, 814, -262, 1786, -21838, -13646, 2738, 35506, -379438, -248366, -9175402, -8651114, -7602538, -5505386, -214704262, -206315654, -189538438, -155984006, -88875142, 45342586, -5932822318, -5395951406, -198413006482

Offset: 0

Alonso del Arte, Sep 30 2018

sign

This sequence shows 2^n is neither a lower bound nor an upper bound for the primorials.

Cf. A000079, A054850, A319852, A319857.

restart;
with(NumberTheory);
a := n -> 2^n-product(ithprime(i), i = 1 .. PrimeCounting(n)):
0, seq(a(n), n = 1 .. 15); # Stefano Spezia, Nov 05 2018

Table[2^n - Times@@Select[Range[n], PrimeQ], {n, 0, 31}]

a(n) = 2^n - prod(k=1, primepi(n), prime(k)); \\ Michel Marcus, Nov 05 2018

a(n) = 2^n - n#, where n# is the product of primes less than or equal to n (A034386).

a(n) = A000079(n) - A034386(n) .

cp's OEIS Frontend

A045716 a(n) is the binary order (A029837) of the n-th primorial number, A002110(n).

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

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A082811 Smallest multiple of n beginning with 2.

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A082791 Smallest k such that k*n begins with 2: a(n) = A082811(n)/n.

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A287010 Triangle T(n,m): floor(log(A002110(n))/log(prime(m))).

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A307133 T(n,m) = number of k <= A002110(n) such that A001221(k) = m, where k is a term in A025487.

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A319880 Difference between 2^n and the product of primes less than or equal to n.

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