A065855 -id:A065855 - cp's OEIS frontend

A077463 Number of primes p such that n < p < 2n-2.

0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 10, 10, 11, 11, 11, 12, 13, 13, 14, 13, 13, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 13, 13, 13, 14, 15, 15, 14, 15, 15, 15, 15, 15

Offset: 1

Eric W. Weisstein, Nov 05 2002

nonn

a(n) > 0 for n > 3 by Bertrand's postulate (and Chebyshev's proof of 1852). - Jonathan Vos Post, Aug 08 2013

			a(19) = 3, the first value smaller than a previous value, because the only primes between 19 and 2 * 19 - 2 = 36 are {23,29,31}. - _Jonathan Vos Post_, Aug 08 2013

J. Sondow and E. Weisstein, Bertrand's Postulate, World of Mathematics
M. Tchebichef, Memoire sur les nombres premiers, J. Math. Pures Appliq. 17 (1852) 366.

Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

a[n_] := PrimePi[2n - 2] - PrimePi[n]; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 31 2012 *)

A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2a(n)+1, a(c_n) = 2a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).

Original entry on oeis.org

1, 3, 7, 2, 15, 6, 5, 14, 4, 30, 31, 12, 13, 10, 28, 8, 11, 60, 29, 62, 24, 26, 9, 20, 56, 16, 22, 120, 61, 58, 63, 124, 48, 52, 18, 40, 25, 112, 32, 44, 27, 240, 21, 122, 116, 126, 57, 248, 96, 104, 36, 80, 17, 50, 224, 64, 88, 54, 23, 480, 121, 42, 244, 232, 252, 114, 59, 496, 192, 208, 125, 72, 49, 160, 34, 100

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

This permutation is otherwise like Katarzyna Matylla's A135141, except that the role of even and odd numbers (or alternatively: primes and composites) has been swapped.
Because 2 is the only even prime, it implies that, apart from a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).
This also implies that for each odd composite (A071904) there exists a separate infinite cycle in this permutation, apart from 9 and 15 which are in the same infinite cycle: (..., 23, 9, 4, 2, 3, 7, 5, 15, 28, 120, 82, 46, ...).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A246378.
Cf. A000720, A010051, A065855, A071904, A246348, A246369, A246370.
Other related or similar permutations: A135141, A054429, A246201, A245703, A246376, A246379, A243071, A246681, A236854.
Differs from A237427 for the first time at n=19, where a(19) = 29, while A237427(19) = 62.

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n)  = 1+(2*a(A000720(n))), otherwise a(n) = 2*a(A065855(n)).
As a composition of related permutations:
a(n) = A054429(A135141(n)).
a(n) = A135141(A236854(n)).
a(n) = A246376(A246379(n)).
a(n) = A246201(A245703(n)).
a(n) = A243071(A246681(n)). [For n >= 1].
Other identities.
For all n > 1 the following holds:
A000035(a(n)) = A010051(n). [Maps primes to odd numbers > 1, and composites to even numbers, in some order. Permutations A246379 & A246681 have the same property].

A376759 Number of composite numbers c with n < c <= 2*n.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 5, 6, 6, 6, 8, 8, 10, 11, 11, 11, 13, 14, 15, 16, 16, 16, 18, 18, 19, 20, 20, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 32, 32, 34, 35, 35, 36, 38, 39, 39, 40, 40, 40, 42, 42, 42, 43, 43, 44, 46, 47, 49, 50, 51, 51, 52, 52, 54, 55, 55, 55, 57, 58, 60, 61, 61, 61, 62, 63, 64, 65, 66, 66, 68, 68, 69, 70, 70, 71, 73, 73, 73, 74, 75, 76, 77, 77

Offset: 1

N. J. A. Sloane, Oct 20 2024

nonn

This completes the set of four: A307912, A376759, A307989, and A075084. Since it is not clear which ones are the most important, and they are easily confused, all four are now in the OEIS.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A376759 := proc(n) chi(2*n) - chi(n); end;
a := [seq(A376759(n),n=1..120)];

Table[PrimePi[n] - PrimePi[2*n] + n, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

from sympy import primepi
def A376759(n): return n+primepi(n)-primepi(n<<1) # Chai Wah Wu, Oct 20 2024

a(n) = A000720(n) - A000720(2*n) + n. - Paolo Xausa, Oct 22 2024

A085970 Number of integers ranging from 2 to n that are not prime-powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 42, 43

Offset: 1

Reinhard Zumkeller, Jul 06 2003

nonn

For n > 2, a(n) gives the number of duplicate eliminations performed by the Sieve of Eratosthenes when sieving the interval [2, n]. - Felix Fröhlich, Dec 10 2016
Number of terms of A024619 <= n. - Felix Fröhlich, Dec 10 2016
First differs from A082997 at n = 30. - Gus Wiseman, Jul 28 2022

			The a(30) = 13 numbers: 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30. - _Gus Wiseman_, Jul 28 2022

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A000720, A024619, A085972.
The complement is counted by A065515, without 1's A025528.
For primes instead of prime-powers we have A065855, with 1's A062298.
Partial sums of A143731.
The version not treating 1 as a prime-power is A356068.
A000688 counts factorizations into prime-powers.
A001222 counts prime-power divisors.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A023893, A023894, A036234, A069513, A207375, A354911, A355743, A356064, A356065.

With[{nn = 75}, Table[n - Count[#, k_ /; k < n] - 1, {n, nn}] &@ Join[{1}, Select[Range@ nn, PrimePowerQ]]] (* Michael De Vlieger, Dec 11 2016 *)

a(n) = my(i=0); forcomposite(c=4, n, if(!isprimepower(c), i++)); i \\ Felix Fröhlich, Dec 10 2016

from sympy import primepi, integer_nthroot
def A085970(n): return n-1-sum(primepi(integer_nthroot(n,k)[0]) for k in range(1,n.bit_length())) # Chai Wah Wu, Aug 20 2024

a(n) = Max{A024619(k)<=n} k;

a(n) = n - A065515(n) = A085972(n) - A000720(n).

Name modified by Gus Wiseman, Jul 28 2022. Normally 1 is not considered a prime-power, cf. A000961, A246655.

A091245 Number of reducible GF(2)[X]-polynomials in range [0,n].

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 56

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

Analogous to A065855.

			In range [0,8] there are the following four reducible polynomials: 4,5,6,8 thus a(8) = 4.

Partial sums of A091247. Cf. A091242.

first(n)=my(s); concat([0,0], vector(n-1,k, s += !polisirreducible(Pol(binary(k+1))*Mod(1,2)))) \\ Charles R Greathouse IV, Sep 02 2015

A246514 Number of composite numbers between prime(n) and 2*prime(n) exclusive.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 12, 14, 17, 22, 23, 27, 31, 33, 37, 41, 45, 48, 53, 56, 59, 63, 67, 72, 77, 80, 83, 87, 90, 94, 103, 107, 111, 113, 121, 124, 128, 134, 138, 144, 148, 150, 158, 160, 164, 166, 175, 184, 188, 190, 193, 199, 201, 209, 214, 219, 226, 228, 234

Offset: 1

Odimar Fabeny, Aug 28 2014

			2 P 4 = 0,
3 4 P 6 = 1,
5 6 P 8 9 10 = 3,
7 8 9 10 P 12 P 14 = 4,
11 12 P 14 15 16 P 18 P 20 21 22 = 7
and so on.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

A246515 := proc(n) local p;  p:=ithprime(n); n - 1 + p - numtheory:-pi(2*p - 1); end; # N. J. A. Sloane, Oct 20 2024
[seq(A246515(n),n=1..120)];

Table[Prime[n] - PrimePi[2*Prime[n]] + n - 1, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

s=[]; forprime(p=2, 1000, n=0; for(q=p+1, 2*p-1, if(!isprime(q), n++)); s=concat(s, n)); s \\ Colin Barker, Aug 28 2014

a(n)=prime(n)+n-1-primepi(2*prime(n))
vector(100, n, a(n)) \\ Faster program. Jens Kruse Andersen, Aug 28 2014

from sympy import prime, primepi
def A246514(n): return (m:=prime(n))+n-1-primepi(m<<1) # Chai Wah Wu, Oct 22 2024

a(n) + A070046(n) = number of numbers between prime(n) and 2*prime(n), which is prime(n)-1. - N. J. A. Sloane, Aug 28 2014

More terms from Colin Barker, Aug 28 2014

A307912 a(n) = n - 1 - pi(2*n-1) + pi(n), where pi is the prime counting function.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 4, 5, 5, 5, 7, 7, 9, 10, 10, 10, 12, 13, 14, 15, 15, 15, 17, 17, 18, 19, 19, 20, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 29, 31, 31, 33, 34, 34, 35, 37, 38, 38, 39, 39, 39, 41, 41, 41, 42, 42, 43, 45, 46, 48, 49, 50, 50, 51, 51, 53, 54

Offset: 1

Wesley Ivan Hurt, May 09 2019

For n > 1, a(n) is the number of composites in the closed interval [n+1, 2n-1].

a(n) is also the number of composites appearing among the largest parts of the partitions of 2n into two distinct parts.

			a(7) = 4; there are 4 composites in the closed interval [8, 13]: 8, 9, 10 and 12.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Index entries for sequences related to partitions

Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A307912 := proc(n) chi(2*n-1) - chi(n); end;
A := [seq(A307912(n),n=1..120)]; # N. J. A. Sloane, Oct 20 2024

Table[n - 1 - PrimePi[2 n - 1] + PrimePi[n], {n, 100}]

from sympy import primepi
def A307912(n): return n+primepi(n)-primepi((n<<1)-1)-1 # Chai Wah Wu, Oct 20 2024

a(n) = n - 1 - A060715(n).

a(n) = n - 1 - A000720(2*n-1) + A000720(n).

A307989 a(n) = n - pi(2*n) + pi(n-1), where pi is the prime counting function.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 6, 6, 6, 7, 8, 9, 11, 11, 11, 12, 14, 14, 16, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 23, 25, 26, 26, 27, 27, 27, 29, 30, 30, 31, 32, 33, 35, 35, 36, 37, 39, 39, 40, 40, 40, 41, 42, 42, 43, 43, 44, 45, 47, 48, 50, 51, 51, 52, 52, 53, 55

Offset: 1

Wesley Ivan Hurt, May 09 2019

a(n) is the number of composites in the closed interval [n, 2n-1].

a(n) is also the number of composites among the largest parts of the partitions of 2n into two parts.

			a(7) = 4; There are 7 partitions of 2*7 = 14 into two parts (13,1), (12,2), (11,3), (10,4), (9,5), (8,6), (7,7). Among the largest parts 12, 10, 9 and 8 are composite, so a(7) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Index entries for sequences related to partitions

Cf. A000720, A035250.
Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A307989 := proc(n) chi(2*n-1) - chi(n-1); end;
a := [seq(A307989(n),n=1..120)];

Table[n - PrimePi[2 n] + PrimePi[n - 1], {n, 100}]

from sympy import primepi
def A307989(n): return n+primepi(n-1)-primepi(n<<1) # Chai Wah Wu, Oct 20 2024

a(n) = n - A035250(n).

a(n) = n - A000720(2*n) + A000720(n-1).

A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

Original entry on oeis.org

1, 3, 8, 23, 29, 33, 45, 98, 153, 188, 216, 262, 281, 366, 428, 589, 737, 1182, 1830, 1878, 2190, 2224, 3076, 3301, 3384, 3426, 3643, 3792, 4521, 4611, 7969, 8027, 8687, 12541, 14356, 14861, 15782, 17005, 19025, 23282, 30801, 31544, 33607, 34201, 34214, 38589

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A073051.

A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of composite numbers begin:
   4
   6
   8   9  10
  12
  14  15  16
  18
  20  21  22
  24  25  26  27  28
  30
  32  33  34  35  36
  38  39  40
  42
  44  45  46
  48  49  50  51  52
  54  55  56  57  58
  60
  62  63  64  65  66
  68  69  70
  72
  74  75  76  77  78
  80  81  82
  84  85  86  87  88
  90  91  92  93  94  95  96
  98  99 100
The a(n)-th rows are:
   4
   8   9  10
  24  25  26  27  28
  90  91  92  93  94  95  96
 114 115 116 117 118 119 120 121 122 123 124 125 126
 140 141 142 143 144 145 146 147 148
 200 201 202 203 204 205 206 207 208 209 210

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The unsorted version is A073051, firsts of A176246.
For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite runs we have the triple (1,2,7), firsts of A373403.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[10000],CompositeQ],#1+1==#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A376760 Let c(n) = A002808(n) denote the n-th composite number; a(n) = number of composite numbers c with c(n) <= c <= 2*c(n).

Original entry on oeis.org

3, 5, 7, 7, 7, 9, 12, 12, 12, 15, 17, 17, 17, 19, 20, 21, 21, 22, 24, 26, 27, 27, 28, 28, 30, 31, 31, 33, 36, 36, 37, 40, 40, 41, 41, 41, 43, 43, 44, 44, 45, 48, 51, 52, 52, 53, 53, 56, 56, 56, 59, 62, 62, 62, 63, 64, 66, 67, 67, 69, 70, 71, 71, 72, 74, 74, 75, 76, 77, 78, 78, 80, 80, 80, 83, 86, 87, 87, 90, 93, 94, 94, 96, 96, 97, 97, 98, 99, 99, 99, 100, 101, 102, 103

Offset: 1

N. J. A. Sloane, Oct 22 2024

nonn

There are three other versions: composite c with c(n) < c < 2*c(n): a(n)-2; c(n) <= c < 2*c(n): a(n) - 1; and c(n) < c <= 2*c(n): also a(n) - 1.

			The 5th composite number is 10, and 10, 12, 14, 15, 16, 18, 20 are composite, so a(5) = 7.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
t := []: for n from 2 to 200000 do if not isprime(n) then t := [op(t), n]; fi; od: # precompute A002808
ithchi := proc(n) t[n]; end: # returns n-th composite number A002808 for any n <= 182015, analogous to ithprime
A376760 := proc(n) chi(2*ithchi(n)) - n + 1; end;
[seq(A376760(n),n=1..120)];

MapIndexed[2*# - PrimePi[2*#] - #2[[1]] &, Select[Range[100], CompositeQ]] (* Paolo Xausa, Oct 22 2024 *)

from sympy import composite, primepi
def A376760(n): return (m:=composite(n)<<1)-primepi(m)-n # Chai Wah Wu, Oct 22 2024

a(n) = 2*A002808(n) - A000720(2*A002808(n)) - n. - Paolo Xausa, Oct 22 2024

cp's OEIS Frontend

A077463 Number of primes p such that n < p < 2n-2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2*a(n)+1, a(c_n) = 2*a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A376759 Number of composite numbers c with n < c <= 2*n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A085970 Number of integers ranging from 2 to n that are not prime-powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A091245 Number of reducible GF(2)[X]-polynomials in range [0,n].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A246514 Number of composite numbers between prime(n) and 2*prime(n) exclusive.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A307912 a(n) = n - 1 - pi(2*n-1) + pi(n), where pi is the prime counting function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2a(n)+1, a(c_n) = 2a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).