A035250 -id:A035250 - cp's OEIS frontend

A087010 Number of primes of form 4*k+1 between n and 2n (inclusive).

0, 0, 1, 1, 1, 0, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 5, 6, 6, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 10, 9, 10, 10, 10, 9, 9

Offset: 1

Jason Earls, Jul 30 2003

nonn

Erdős proved that between any n > 7 and its double there are always at least two primes, one of form 4*k+1 and one of form 4*k+3.

B. Schechter, "My Brain is Open: The Mathematical Journeys of Paul Erdős," Simon & Schuster, New York, 1998, p. 62.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A035250, A087011, A087012.

[#[p:p in PrimesInInterval(n,2*n)| p mod 4 eq 1]:n in [1..110]]; // Marius A. Burtea, Dec 16 2019

a[n_] := Module[{c = 0}, Do[If[Mod[k, 4] == 1 && PrimeQ[k], c++], {k, n, 2 n}]; c]; Array[a, 100] (* Amiram Eldar, Dec 16 2019 *)

A087011 Number of primes of form 4*k+3 between n and 2n (inclusive).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 7, 7, 8, 8, 7, 7, 8, 8, 7, 7, 7, 7, 8, 8, 8, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11

Offset: 1

Jason Earls, Jul 30 2003

nonn

Erdős proved that between any n > 7 and its double there are always at least two primes, one of form 4*k+1 and one of form 4*k+3.

B. Schechter, "My Brain is Open: The Mathematical Journeys of Paul Erdős," Simon & Schuster, New York, 1998, p. 62.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A035250, A087010, A087012.

[#[p:p in PrimesInInterval(n,2*n)| p mod 4 eq 3]:n in [1..100]]; // Marius A. Burtea, Dec 16 2019

a[n_] := Module[{c = 0}, Do[If[Mod[k, 4] == 3 && PrimeQ[k], c++], {k, n, 2 n}]; c]; Array[a, 100] (* Amiram Eldar, Dec 16 2019 *)

A087012 Numbers m such that the number of primes of form 4k+1 between m and 2m equals the number of primes of form 4k+3 between m and 2m (inclusive).

Original entry on oeis.org

1, 3, 4, 5, 8, 10, 11, 12, 13, 15, 20, 22, 23, 24, 25, 26, 31, 34, 35, 37, 49, 50, 52, 53, 57, 58, 59, 62, 63, 69, 72, 73, 75, 79, 82, 83, 84, 85, 86, 91, 92, 93, 94, 95, 97, 99, 141, 147, 148, 149, 152, 153, 164, 165, 168, 175, 176, 182, 183, 187, 188, 189, 200, 244, 245

Offset: 1

Jason Earls, Jul 30 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A035250, A087010, A087011.

f:=func; [k:k in [1..250]|f(k,1) eq f(k,3)]; // Marius A. Burtea, Dec 16 2019

seqQ[n_] := Module[{c1 = 0, c3 = 0}, Do[If[Mod[k, 4] == 1 && PrimeQ[k], c1++]; If[Mod[k, 4] == 3 && PrimeQ[k], c3++], {k, n, 2 n}]; c1 == c3]; Select[Range[250], seqQ] (* Amiram Eldar, Dec 16 2019 *)
npfQ[n_]:=With[{prs=Select[Range[n,2n],PrimeQ]},Length[Select[prs,Mod[#,4]==1&]]==Length[Select[prs,Mod[#,4]==3&]]]; Select[ Range[ 250],npfQ] (* Harvey P. Dale, Sep 25 2024 *)

for(m=1,250,my(k1=0,k3=0);forprime(p=m,2*m,if(p%4==1,k1++);if(p%4==3,k3++));if(k1==k3,print1(m," "))) \\ Hugo Pfoertner, Dec 16 2019

A088018 Number of twin-prime pairs between n and 2n (inclusive).

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 6, 6, 6, 6

Offset: 1

T. D. Noe, Sep 18 2003, Feb 17 2011

To be counted, both members of the twin-prime pair must be between n and 2n, inclusive. It appears that a(n) > 0 for all n > 6. However, it has not been proved that there are an infinite number of twin primes.

Same as the number of lower twin primes between n-1 and 2(n-1), exclusive. If the twin prime conjecture is true, there are at least n lower twin primes between x/2 and x for all x >= A186312(n).

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A035250 (number of primes between n and 2n), A088019 (number of twin primes between n and 2n).

nn=100; p=Select[Prime[Range[PrimePi[2*nn]]], PrimeQ[#+2] &]; t=Table[0, {nn}]; Do[t[[Span[Ceiling[i/2], Min[nn,i-1]]]]++, {i, p}]; Prepend[t,0]
Table[Total[Length /@ Split[Select[Range[n, 2 n], PrimeQ], #2 - #1 == 2 &] - 1], {n, 105}] (* Jayanta Basu, Aug 12 2013 *)

A088019 Number of twin primes between n and 2n (inclusive).

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Sep 18 2003

Here a twin prime is counted even if only one member of the twin-prime pair is between n and 2n, inclusive. Note that this sequence is very close to 2*A088018. It appears that a(n) > 0 for all n > 1. However, it has not been proved that there are an infinite number of twin primes.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A035250 (number of primes between n and 2n), A088018 (number of twin-prime pairs between n and 2n).

pl=Prime[Range[PrimePi[20000]]]; twl={}; Do[If[pl[[i-1]]+2==pl[[i]], twl=Join[twl, {pl[[i-1]], pl[[i]]}]], {i, 2, Length[pl]}]; twl=Union[twl]; i1=1; i2=1; nMin=(twl[[1]]-1)/2; nMax=(twl[[ -1]]+1)/2; Join[Table[0, {nMin-1}], Table[While[twl[[i1]]

A352777 a(n) = Sum_{p <= n <= q < 2n, p,q prime} (p * q).

Original entry on oeis.org

0, 10, 40, 60, 120, 180, 527, 408, 697, 1020, 1680, 2016, 2952, 2419, 3608, 4879, 6902, 5916, 10703, 9240, 12397, 15708, 20400, 22800, 22800, 22800, 28100, 28100, 36249, 40119, 59520, 54560, 54560, 65280, 65280, 76640, 108744, 101455, 101455, 117018, 141372, 151368, 178716

Offset: 1

Wesley Ivan Hurt, Apr 02 2022

nonn

Total area of all unique p X q rectangles with p,q prime such that p <= n <= q < 2n.

			a(5) = 120; the 6 unique p X q rectangles, with p,q prime such that p <= 5 <= q < 10 are: 2 X 5, 2 X 7, 3 X 5, 3 X 7, 5 X 5, and 5 X 7. The total area of all rectangles is 2*5 + 2*7 + 3*5 + 3*7 + 5*5 + 5*7 = 120.

Cf. A000720 (pi), A035250, A352753, A352754, A352749.

Table[Sum[Sum[k (2 n - i) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, n}], {i, n}], {n, 100}]

A143537 Triangle read by rows: T(n,k) = number of primes in the interval [k..n], n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 1, 0, 3, 3, 2, 1, 1, 3, 3, 2, 1, 1, 0, 4, 4, 3, 2, 2, 1, 1, 4, 4, 3, 2, 2, 1, 1, 0, 4, 4, 3, 2, 2, 1, 1, 0, 0, 4, 4, 3, 2, 2, 1, 1, 0, 0, 0, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1, 0, 6, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Aug 23 2008

Old name: triangle read by rows, A000012 * A143536, 1<=k<=n.

			Triangle T(n,k) begins:
n\k 1  2  3  4  5  6  7  8 ...
1:  0;
2:  1, 1;
3:  2, 2, 1;
4:  2, 2, 1, 0;
5:  3, 3, 2, 1, 1;
6:  3, 3, 2, 1, 1, 0;
7:  4, 4, 3, 2, 2, 1, 1;
8:  4, 4, 3, 2, 2, 1, 1, 0;
...

Row sums are A034387.
Column k=1 gives A000720.
Main diagonal gives A010051.
T(2n,n) gives A035250.
Cf. A143536.

T(n,k) = pi(n) - pi(k-1), where pi = A000720. - Ilya Gutkovskiy, Mar 19 2020

New name and corrected by Ilya Gutkovskiy, Mar 19 2020

A220850 a(n+1) is equal to a(n) plus the number of primes between a(n) and 2*a(n) inclusively.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 17, 22, 28, 35, 43, 53, 65, 78, 93, 111, 129, 153, 179, 210, 245, 285, 328, 381, 441, 508, 582, 668, 764, 870, 990, 1123, 1270, 1436, 1625, 1825, 2054, 2309, 2590, 2904, 3246, 3631, 4052, 4512, 5022, 5582, 6197, 6872, 7612, 8421, 9312

Offset: 1

Robert G. Wilson v, Dec 22 2012

			a(6) = the number of primes between a(5) and 2*a(5) plus a(5) = the number of primes [8, 16] + 8 = 2 + 8 = 10.

Robert G. Wilson v, Table of n, a(n) for n = 1..558

Cf. A035250, A220851, inspired by A084140.

f[n_] := PrimePi[ 2n] - PrimePi[n - 1]; NestList[# +f@# &, 1, 50]

A226983 a(n) = ceiling(n/2) - pi(2n) + pi(n-1).

Original entry on oeis.org

0, -1, 0, 0, 1, 1, 1, 2, 2, 1, 2, 2, 3, 4, 4, 3, 4, 5, 5, 6, 6, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 10, 9, 10, 9, 9, 10, 11, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15, 15, 15, 14, 15, 15, 15, 15, 15, 15, 16, 17, 18, 19, 20, 19, 20, 19, 20, 21, 21, 20, 21, 22, 23, 24

Offset: 1

Wesley Ivan Hurt, Jun 25 2013

The number of partitions of 2n into exactly two parts such that the first part is an odd composite integer, n > 2.

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions

Cf. A000720, A004526, A035250.

with(numtheory); seq(ceil(k/2)-(pi(2*k)-pi(k-1)),k=1..100);

Table[Floor[(n + 1) / 2] - (PrimePi[2 n] - PrimePi[n - 1]), {n, 100}] (* Vincenzo Librandi, Dec 07 2016 *)

a226983(n) = if(n==1, 0, ceil(n/2) - primepi(2*n) + primepi(n-1)) \\ Michael B. Porter, Jun 29 2013

a(n) = floor((n+1)/2) - ( pi(2n) - pi(n-1) ) = A004526(n+1) - A035250(n).

A284437 Number of primes between n and 2^n inclusive.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 15, 28, 50, 93, 168, 305, 559, 1023, 1894, 3506, 6536, 12245, 22993, 43383, 82017, 155603, 295939, 564155, 1077862, 2063680, 3957800, 7603544, 14630834, 28192741, 54400018, 105097555, 203280210, 393615795, 762939100, 1480206268, 2874398504, 5586502337

Offset: 0

Vincenzo Librandi, Mar 27 2017

nonn

			a(0) = 0 because there are 0 primes between 0 and 2^0.
a(5) = 9 because there are 9 primes between 5 and 2^5: 5, 7, 11, 13, 17, 19, 23, 29, 31 (we count the boundary of the interval in this case).

Amiram Eldar, Table of n, a(n) for n = 0..92 (terms 0..47 from Vincenzo Librandi)

Cf. A000720, A007053, A035250, A060715, A080339, A284275.

[0] cat [#PrimesInInterval(n, 2^n): n in [1..28]];

Join[{0}, f[n_]:=PrimePi[2^n] - PrimePi[n-1]; Array[f, 37]]

a(n) = A284275(n) + A080339(n) for n >= 1. - Amiram Eldar, Jun 11 2024

cp's OEIS Frontend

A087010 Number of primes of form 4*k+1 between n and 2n (inclusive).

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A087011 Number of primes of form 4*k+3 between n and 2n (inclusive).

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A087012 Numbers m such that the number of primes of form 4*k+1 between m and 2*m equals the number of primes of form 4*k+3 between m and 2*m (inclusive).

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A088018 Number of twin-prime pairs between n and 2n (inclusive).

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A088019 Number of twin primes between n and 2n (inclusive).

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A352777 a(n) = Sum_{p <= n <= q < 2n, p,q prime} (p * q).

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A143537 Triangle read by rows: T(n,k) = number of primes in the interval [k..n], n >= 1, 1 <= k <= n.

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A220850 a(n+1) is equal to a(n) plus the number of primes between a(n) and 2*a(n) inclusively.

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A226983 a(n) = ceiling(n/2) - pi(2n) + pi(n-1).

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A087012 Numbers m such that the number of primes of form 4k+1 between m and 2m equals the number of primes of form 4k+3 between m and 2m (inclusive).