A108954 -id:A108954 - 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 *)

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

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).

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

A376761 Number of primes between the n-th composite number c(n) and 2*c(n).

Original entry on oeis.org

2, 2, 2, 3, 4, 4, 3, 4, 5, 4, 4, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 9, 10, 10, 9, 10, 10, 11, 12, 12, 13, 13, 14, 14, 13, 12, 12, 13, 13, 14, 13, 14, 15, 14, 13, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 20, 21, 20, 19, 19, 20, 19, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 23, 23, 23, 23, 23, 24, 23, 24, 24, 24, 24, 24, 25

Offset: 1

N. J. A. Sloane, Oct 22 2024

nonn

Obviously the endpoints are not counted (since they are composite).

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.

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

from sympy import composite, primepi
def A376761(n): return n+1-(m:=composite(n))+primepi(m<<1) # Chai Wah Wu, Oct 22 2024

a(n) = A000720(2*A002808(n)) - A002808(n) + n + 1. - Paolo Xausa, Oct 22 2024

A084140 a(n) is the smallest number j such that if x >= j there are at least n primes between x and 2x exclusively.

Original entry on oeis.org

2, 6, 9, 15, 21, 24, 30, 34, 36, 49, 51, 54, 64, 75, 76, 84, 90, 91, 114, 115, 117, 120, 121, 132, 135, 141, 154, 156, 174, 175, 184, 187, 201, 205, 210, 216, 217, 220, 231, 244, 246, 252, 285, 286, 294, 297, 300, 301, 304, 321, 322, 324, 327, 330, 339, 360, 364

Offset: 1

Harry J. Smith, May 15 2003

nonn

For all m >= a(n) there are at least n primes between m and 2m exclusively. This calculation relies on the fact that pi(2m) - pi(m) > m/(3*log(m)) for m >= 5. This is one more than the terms of A084139 with offset changed from 0 to 1.
For n > 5889, pi(2n) - pi(n) > f(2, 2n) - f(3, n) where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). This may be useful for checking larger terms. The constant 3 can be improved at the cost of an increase in the constant 5889. - Charles R Greathouse IV, May 02 2012
A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime >= x. - John W. Nicholson, Dec 21 2012
a(1) = ceiling((A104272(1)+1)/2) modifies the only even prime, 2; which has been stated, in Formula, as a(1) = A104272(1); for all others, a(n) = (A104272(n)+1)/2 = ceiling ((A104272(n)+1)/2). - John W. Nicholson, Dec 24 2012
Srinivasan's Lemma (2014): previousprime(a(n)) = p_(k-n) < (p_k)/2, where the n-th Ramanujan Prime R_n is the k-th prime p_k, and with n > 1. Proof: By the minimality of R_n, the interval ((p_k)/2,p_k] contains exactly n primes, so p_(k-n) < (p_k)/2. - Copied and adapted from a comment by Jonathan Sondow in A168421 by John W. Nicholson, Feb 17 2015

			a(11)=51 since there are at least 11 primes between m and 2m for all m >= 51 and this is not true for any m < 51. Although a(100)=720 is not listed, for all m >= 720, there are at least 100 primes between m and 2m.

Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, 1991, p. 140.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag, 2004, p. 181.

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv version, arXiv:1212.2785 [math.NT], 2012.
Anitha Srinivasan, An upper bound for Ramanujan primes, Integers, 19 (2014), #A19.
Eric Weisstein's World of Mathematics, Bertrand's Postulate.

Cf. A108954, A060715, A060756, A084138, A084139, A084141, A084142, A168421, A168425.

nn = 60;
R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
Ceiling[(A104272 + 1)/2] (* Jean-François Alcover, Nov 07 2018, after T. D. Noe in A104272 *)

a(1) = A104272(1); for n >= 2, a(n) = (A104272(n)+1)/2. - Vladimir Shevelev, Dec 07 2012

a(n) = ceiling((A104272(n)+1)/2) for n >= 1. - John W. Nicholson, Dec 24 2012

A263995 Cardinality of the union of the set of sums and the set of products made from pairs of integers from {1..n}.

Original entry on oeis.org

2, 4, 7, 11, 15, 20, 27, 32, 39, 46, 56, 63, 75, 83, 93, 102, 118, 127, 146, 156, 169, 182, 204, 215, 231, 245, 261, 274, 302, 315, 346, 361, 379, 398, 418, 432, 469, 489, 510, 527, 567, 585, 627, 647, 669, 693, 739, 756, 788, 810, 838, 862, 914, 937

Offset: 1

Hugo Pfoertner, Nov 15 2015

nonn

The November 2015 - Feb 2016 round of Al Zimmermann's Programming Contests asks for sets of positive integers (instead of {1..n}) minimizing the cardinality of the union of the sum-set and the product-set for set sizes 40, 80, ..., 1000. [corrected by Al Zimmermann, Nov 24 2015]

			a(3)=7 because the union of the set of sums {1+1, 1+2, 1+3, 2+2, 2+3, 3+3} and the set of products {1*1, 1*2, 1*3, 2*2, 2*3, 3*3} = {2,3,4,5,6} U {1,2,3,4,6,9} = {1,2,3,4,5,6,9} has cardinality 7.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag New York, 2004. Problem F18.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
P. Erdős and E. Szemeredi, On sums and products of integers, Studies in Pure Mathematics, Birkhäuser, Basel, 1983, pp. 213-218. DOI:10.1007/978-3-0348-5438-2_19
Al Zimmermann's Programming Contests, Sums and Products I, Nov 2015 - Feb 2016.

Cf. A027424, A108954, A263996.

a(n) = {my(v = [1..n]); v = setunion(setbinop((x,y)->(x+y), v), setbinop((x,y)->(x*y), v)); #v;} \\ Michel Marcus, Apr 13 2022

from math import prod
from itertools import combinations_with_replacement
def A263995(n): return len(set(sum(x) for x in combinations_with_replacement(range(1,n+1),2)) | set(prod(x) for x in combinations_with_replacement(range(1,n+1),2))) # Chai Wah Wu, Apr 15 2022

a(n) = A027424(n) + A108954(n). - Jon Maiga, Jan 03 2022

A074990 Number of primes in the interval (n,3n].

Original entry on oeis.org

2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 8, 9, 8, 9, 8, 9, 10, 10, 10, 11, 12, 12, 13, 14, 13, 14, 13, 13, 14, 15, 16, 17, 17, 18, 18, 18, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 21, 21, 21, 21, 22, 23, 23, 24, 23, 24, 24, 24, 24, 25, 26, 27, 27, 27, 27, 27, 27, 27, 26, 26, 27, 28

Offset: 1

Zak Seidov, Oct 02 2002

nonn

Differences a(n+1)-a(n) are 0, 1 or -1.

			a(4) = 3 because between 4 and 12 there are 3 primes: 5, 7, 11; a(7) = 4 because between 7 and 21 there are 4 primes: 11, 13, 17, 19.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
M. Das, G. Paul, Generalization of Bertrand's Postulate and Upper Bounds on the Number of Primes in Certain Intervals, arXiv:1710.09891 [math.NT], 2017.

Cf. A000040, A000720, A010051, A108954, A075699.

A074990 := proc(n)
    numtheory[pi](3*n)-numtheory[pi](n) ;
end proc: # R. J. Mathar, Nov 03 2017

Mathematica
```
s=3; a[n_] := PrimePi[s*n]-PrimePi[n]
```

a(n) = A000720(3n) - A000720(n).

a(n) = Sum_{j=n+1..3*n} A010051(j).

cp's OEIS Frontend

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

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

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

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

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

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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).

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A376761 Number of primes between the n-th composite number c(n) and 2*c(n).

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