A063124 -id:A063124 - cp's OEIS frontend

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

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

A376754 Length of n-th run of primes in A376198.

Original entry on oeis.org

2, 3, 4, 8, 13, 24, 43, 78, 142, 261, 479, 894, 1674, 3118, 5873, 11102, 20992, 39830, 75906, 144652, 276720, 529865, 1016535, 1954167, 3761091, 7250277, 13993031, 27042169, 52313384, 101320082, 196422988, 381154209, 740280217, 1438969498, 2799310690, 5449726356

Offset: 1

N. J. A. Sloane, Oct 03 2024

nonn

Theorem: a(n) is the number of steps needed for the prime A376751(n) to "double" in the normal sequence of primes. More precisely, if A376751(n) = prime(j), then a(n) = A063124(j). For example, A376751(8) = 521 = prime(98), and A063124(98) = 78 = a(8). (The result seems to be off by 1 at n = 4, for reasons I don't understand yet.) - N. J. A. Sloane, Oct 04 2024

Cf. A376198-A376201, A376750-A376753.

A378746 a(n) = Product{i=n..primepi(2*prime(n))} prime(i).

Original entry on oeis.org

6, 15, 35, 1001, 46189, 96577, 6678671, 14535931, 1348781387, 146078888479, 18128893780549, 203079283326684719, 433601713048867373, 877779077635511999, 1816798556036292277, 39006703653387621491281, 969956148531489825059765363, 16439934720872708899318057, 4483790064773102589474664169, 1274400211992152128527851190601

Offset: 1

Antti Karttunen, Dec 08 2024

nonn

Each a(n) is a proper divisor of A007741(n).
Each a(n) or one of its proper divisors is present in A337372.
a(n) is the least product of consecutive primes starting from A000040(n) such that A003961(a(n)) > 2*a(n). Note that 6, 15, 35, 46189 and 96577 are all present in A337372, i.e., are primitively primeshift-abundant numbers. For 1001 = 7*11*31, it is however its proper divisor 91 = 7*31 which gets that honor.

Antti Karttunen, Table of n, a(n) for n = 1..344

Cf. A003961, A007741, A063124 [= A001222(a(n))], A337372.
Subsequence of A097889.
Cf. also A378745.

A378746(n) = prod(i=n,primepi(2*prime(n)),prime(i));

A124136 The list of primes p such that the number of primes in the open interval (p,2p) is larger than the number of primes in the open interval (q,2q) for all q

Original entry on oeis.org

2, 7, 11, 17, 23, 29, 31, 37, 53, 59, 71, 79, 89, 97, 101, 127, 137, 149, 157, 179, 191, 211, 223, 233, 251, 257, 263, 293, 307, 311, 331, 347, 367, 373, 379, 389, 409, 419, 431, 443, 457, 479, 487, 499, 521, 541, 547, 557, 563, 587, 599, 613, 617, 631, 641

Offset: 1

Jani Melik, Nov 30 2006

nonn

Sequence A060715(n) lists the number of primes in the open interval (n,2*n).
If we extract its sublist for n a prime, the number of primes in the open interval (p,2*p), we have A070046(m) = 1, 1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 9, 9, 9, 11, 13, 12, 13, 14, 13, 15, 15, 16... for the primes p=2, 3, 5, 7, 11, 13, 17, 19 etc.
This sequence lists the primes p = prime(m) that set a new record in A070046(m).
Alternative definition: primes p defined by positions of records in A063124.

			a(1)=prime(1)=2 with 1 prime in the interval (2,4). a(2) is neither 3 (with 1 prime in the interval (3,6)), nor 5 (with 1 prime in the interval (5,10)), but a(2)=7 with 2 primes in the interval (7,14).
The primes 41, 43 and 47 are not in the list because the intervals (41,82), (43,86) and (47,94) contain 9 primes, but the interval (37,74) with the smaller prime p=37 already contained 9 primes.
The prime 53 is in the list because the interval (53,106) contains 11 primes and the intervals (q,2*q) for primes q =2,3, 5, ..,47 contained 9 or less primes.

Cf. A060756, A060715, A084139.

ts_c:=proc(n) local i,j,st_p,max_stp,ans; ans:= [ ]: st_p:=0: max_stp:=0: for i from 2 to n do for j from i+1 to 2*i-1 do if (isprime(j) = 'true') then st_p:=st_p+1: fi od: if (st_p > max_stp and isprime(i) = 'true') then max_stp := st_p: ans:=[ op(ans),i ]: fi; st_p:=0: od: RETURN(ans) end: ts_c(1200);

Definition recovered from the Maple program. - R. J. Mathar, May 21 2025

A376764 Length of n-th region in A376198.

Original entry on oeis.org

3, 8, 15, 33, 63, 132, 260, 528, 1057, 2123, 4236, 8464, 16932, 33860, 67732, 135453, 270927, 541836, 1083716, 2167408, 4334796, 8669604, 17339208, 34678448, 69356872, 138713720, 277427448, 554854896, 1109709804, 2219419612, 4438839324, 8877678536, 17755357084, 35510714180, 71021428384, 142042856736

Offset: 1

N. J. A. Sloane, Oct 30 2024

nonn

We define a region in A376198 to consist of a maximal string of non-prime terms followed by a string of prime terms.

			The first two regions in A376198 consist of terms 1 to 3 (of length a(1) = 3) and terms 4 to 11 (of length a(2) = 8).

Cf. A063124, A376198, A376752, A376763.

Apart from the initial 3, this is the first differences of A376752.

This is also, very approximately, the sum of A376763 and a subsequence of A063124.

cp's OEIS Frontend

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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A376754 Length of n-th run of primes in A376198.

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A124136 The list of primes p such that the number of primes in the open interval (p,2p) is larger than the number of primes in the open interval (q,2q) for all q