A128076 -id:A128076 - cp's OEIS frontend

A131413 Triangle read by rows: A002024 + A128076 - A000012 as infinite lower triangular matrices.

1, 4, 3, 7, 6, 5, 10, 9, 8, 7, 13, 12, 11, 10, 9, 16, 15, 14, 13, 12, 11, 19, 18, 17, 16, 15, 14, 13, 22, 21, 20, 19, 18, 17, 16, 15, 25, 24, 23, 22, 21, 20, 19, 18, 17, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23

Offset: 0

Gary W. Adamson, Jul 08 2007

Row sums = A000566, the heptagonal numbers: (1, 7, 18, 34, 55, ...).

			First few rows of the triangle:
   1;
   4,  3;
   7,  6,  5;
  10,  9,  8,  7;
  13, 12, 11, 10,  9;
  16, 15, 14, 13, 12, 11;
  19, 18, 17, 16, 15, 14, 13;
  ...

Cf. A000566, A002024, A128076.

By rows, (n+1) terms of 3n+1, 3n, 3n-1, ...

a(5) = 5, a(30) = 20 corrected and more terms from Georg Fischer, Jun 07 2023

A060715 Number of primes between n and 2n exclusive.

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, Apr 25 2001

See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
a(A060756(n)) = n and a(m) <> n for m < A060756(n). - Reinhard Zumkeller, Jan 08 2012
For prime n conjecturally a(n) = A226859(n). - Vladimir Shevelev, Jun 27 2013
The number of partitions of 2n+2 into exactly two parts where the first part is a prime strictly less than 2n+1. - Wesley Ivan Hurt, Aug 21 2013

			a(35)=8 since eight consecutive primes (37,41,43,47,53,59,61,67) are located between 35 and 70.

M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer NY 2001.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from T. D. Noe]
C. K. Caldwell, The Prime Glossary, Bertrand's postulate
R. Chapman, Bertrand postulate [Broken link]
Math Olympiads, Bertrand's Postulate [Broken link]
S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
Vladimir Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, arXiv:0909.0715v13 [math.NT]
Vladimir Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4
M. Slone, PlanetMath.org, Proof of Bertrand's conjecture
Jonathan Sondow and Eric Weisstein, Bertrand's Postulate, World of Mathematics
Wikipedia, Proof of Bertrand's postulate
Dr. Wilkinson, The Math Forum, Erdos' Proof
Wolfram Research, Bertrand hypothesis

Cf. A060756, A070046, A006992, A051501, A035250, A101909.
Cf. A000720, A014085, A104272, A143227, A128076.
Cf. A143223, A143224, A143225, A143226.
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.

a060715 n = sum $ map a010051 [n+1..2*n-1]  -- Reinhard Zumkeller, Jan 08 2012

[0] cat [#PrimesInInterval(n+1, 2*n-1): n in [2..80]]; // Bruno Berselli, Sep 05 2012

a := proc(n) local counter, i; counter := 0; from i from n+1 to 2*n-1 do if isprime(i) then counter := counter +1; fi; od; return counter; end:
with(numtheory); seq(pi(2*k-1)-pi(k),k=1..100); # Wesley Ivan Hurt, Aug 21 2013

a[n_]:=PrimePi[2n-1]-PrimePi[n]; Table[a[n],{n,1,84}] (* Jean-François Alcover, Mar 20 2011 *)

{ for (n=1, 1000, write("b060715.txt", n, " ", primepi(2*n - 1) - primepi(n)); ) } \\ Harry J. Smith, Jul 10 2009

from sympy import primerange as pr
def A060715(n): return len(list(pr(n+1, 2*n))) # Karl-Heinz Hofmann, May 05 2022

a(n) = Sum_{k=1..n-1} A010051(n+k). - Reinhard Zumkeller, Dec 03 2009
a(n) = pi(2n-1) - pi(n). - Wesley Ivan Hurt, Aug 21 2013
a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} A010051(A128076(k)). - Wesley Ivan Hurt, Jan 08 2022

Corrected by Dug Eichelberger (dug(AT)mit.edu), Jun 04 2001

More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001

A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 5, 6, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 9, 10, 9, 9, 10, 10, 9, 9, 10, 10, 11, 12, 11, 12, 13, 13, 14, 14, 13, 13, 12, 12, 12, 13, 13, 14, 13, 13, 14, 15, 14, 14, 13, 13, 14, 15, 15, 15, 15, 15, 15, 16, 15, 16

Offset: 1

Cino Hilliard, Jul 22 2005

a(n) < log(4)*n/log(n) < 7*n/(5*log(n)) for n > 1. - Reinhard Zumkeller, Mar 04 2008
Bertrand's postulate is equivalent to the formula a(n) >= 1 for all positive integers n. - Jonathan Vos Post, Jul 30 2008
Number of distinct prime factors > n of binomial(2*n,n). - T. D. Noe, Aug 18 2011
f(2, 2n) - f(3, n) < a(n) < f(3, 2n) - f(2, n) for n > 5889 where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). The constant 3 can be improved. - Charles R Greathouse IV, May 02 2012
For n >= 2, a(n) is the number of primes appearing in the 2nd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

F. Irschebeck, Einladung zur Zahlentheorie, BI Wissenschaftsverlag 1992, p. 40.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 181-182.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tsutomu Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.

Cf. A000720, A060715.
Cf. A067434 (number of prime factors in binomial(2*n,n)), A193990, A074990.
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.

A108954 := proc(n)
    numtheory[pi](2*n)-numtheory[pi](n) ;
end proc: # R. J. Mathar, Nov 03 2017

Table[Length[Select[Transpose[FactorInteger[Binomial[2 n, n]]][[1]], # > n &]], {n, 100}] (* T. D. Noe, Aug 18 2011 *)
f[n_] := Length@ Select[ Range[n + 1, 2n], PrimeQ]; Array[f, 100] (* Robert G. Wilson v, Mar 20 2012 *)
Table[PrimePi[2n]-PrimePi[n],{n,90}] (* Harvey P. Dale, Mar 11 2013 *)

g(n) = for(x=1,n,y=primepi(2*x)-primepi(x);print1(y","))

from sympy import primepi
def A108954(n): return primepi(n<<1)-primepi(n) # Chai Wah Wu, Aug 19 2024

a(n) = A000720(2*n)-A000720(n).
For n > 1, a(n) = A060715(n). - David Wasserman, Nov 04 2005
Conjecture: G.f.: Sum_{i>0} Sum_{j>=i|i+j is prime} x^j. - Benedict W. J. Irwin, Mar 31 2017
From Wesley Ivan Hurt, Sep 20 2021: (Start)
a(n) = Sum_{k=1..n} A010051(2*n-k+1).
a(n) = Sum_{k=n*(n+1)/2+2..(n+1)*(n+2)/2} A010051(A128076(k)). (End)

A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 24 2004

All numbers m occur ceiling(m/2) times, see A004526.

The LCM of the n-th row is A076100. - Michel Marcus, Mar 18 2018

			Triangle begins:
  1;
  2,  3;
  3,  4,  5;
  4,  5,  6,  7;
  5,  6,  7,  8,  9;
  6,  7,  8,  9, 10, 11;
  7,  8,  9, 10, 11, 12, 13;
  8,  9, 10, 11, 12, 13, 14, 15;
  9, 10, 11, 12, 13, 14, 15, 16, 17;
  ... - _Philippe Deléham_, Mar 30 2013

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Bruno Berselli, Illustration of the initial terms.
László Németh, On the Binomial Interpolated Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.8.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002024, A002260, A003056, A004526, A004736, A005408, A005843.
Cf. A016777, A016813, A016861, A037213, A076100, A078112, A093005.
Cf. A094728, A214604, A214661, A319556.
Cf. A128076 (rows reversed).

a094727 n k = n + k
a094727_row n = a094727_tabl !! (n-1)
a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]
-- Reinhard Zumkeller, Jul 22 2012

z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];

Table[n + Range[0, n-1], {n, 12}]//Flatten (* Michael De Vlieger, Dec 16 2016 *)

from math import isqrt
def A094727(n): return ((a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-a)>>1)+n-1 # Chai Wah Wu, Jun 19 2025

flatten([[n+k for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n+1, k) = T(n, k) + 1 = T(n, k+1); T(n+1, k+1) = T(n, k) + 2.
T(n, n - A005843(k)) = A005843(n-k) for 0 <= k <= n/2.
T(n, n - A005408(k)) = A005408(n-k) for 0 <= k < n/2.
T(A005408(n), n) = A016777(n), n >= 0.
Sum_{k=1..n} T(n, k) = A000326(n) (row sums).
T(n, k) = A002024(n,k) + A002260(n,k) - 1. - Reinhard Zumkeller, Apr 27 2006
As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m*(m-3)/2 - 1. - Carl R. White, Jul 30 2009
T(n, k) = n+k-1, n >= k >= 1. - Vincenzo Librandi, Nov 23 2009 [corrected by Klaus Brockhaus, Nov 23 2009]
T(n,k) = A037213((A214604(n,k) + A214661(n,k)) / 2). - Reinhard Zumkeller, Jul 25 2012
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = A002260(n) + A003056(n).
a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
From G. C. Greubel, Mar 10 2024: (Start)
T(3*n-3, n) = A016813(n-1).
T(4*n-4, n) = A016861(n-1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).
Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

A131914 3A002024 - 2A051340.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 10, 8, 6, 4, 13, 11, 9, 7, 5, 16, 14, 12, 10, 8, 6, 19, 17, 15, 13, 11, 9, 7, 22, 20, 18, 16, 14, 12, 10, 8, 25, 23, 21, 19, 17, 15, 13, 11, 9, 28, 26, 24, 22, 20, 18, 16, 14, 12, 10

Offset: 1

Gary W. Adamson, Jul 27 2007

Row sums = the hexagonal numbers, A000384: (1, 6, 15, 28, 45, ...).
From Boris Putievskiy, Jan 24 2013: (Start)
Table T(n,k) = n + 3*k - 3, n, k > 0, read by antidiagonals. General case A209304. Let m be a positive integer. The first column of the table T(n,1) is the sequence of the positive integers A000027. Every subsequent column is formed from the previous column, shifted by m elements.
For m=0 the result is A002260,
for m=1 the result is A002024,
for m=2 the result is A128076,
for m=3 the result is A131914,
for m=4 the result is A209304. (End)

			First few rows of the triangle:
   1;
   4,  2;
   7,  5,  3;
  10,  8,  6,  4;
  13, 11,  9,  7,  5;
  16, 14, 12, 10,  8,  6;
  19, 17, 15, 13, 11,  9,  7;
  ...

Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A002024, A051340, A000384, A003056, A002260, A002024, A128076, A209304.

3*A002024 - 2*A051340 as infinite lower triangular matrices.
From Boris Putievskiy, Jan 24 2013: (Start)
For the general case
a(n) = m*A003056 - (m-1)*A002260.
a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n), where t = floor((-1+sqrt(8*n-7))/2).
For m = 3,
a(n) = 3*A003056 - 2*A002260.
a(n) = 3*(t+1) + 2*(t*(t+1)/2-n), where t = floor((-1+sqrt(8*n-7))/2). (End)

A209304 Table T(n,k)=n+4*k-4 n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 2, 9, 6, 3, 13, 10, 7, 4, 17, 14, 11, 8, 5, 21, 18, 15, 12, 9, 6, 25, 22, 19, 16, 13, 10, 7, 29, 26, 23, 20, 17, 14, 11, 8, 33, 30, 27, 24, 21, 18, 15, 12, 9, 37, 34, 31, 28, 25, 22, 19, 16, 13, 10, 41, 38, 35, 32, 29, 26, 23, 20, 17, 14, 11, 45, 42, 39, 36, 33, 30, 27

Offset: 1

Boris Putievskiy, Jan 18 2013

In general, let m be natural number. The first column of the table T(n,1) is the sequence of the natural numbers A000027. Every next column is formed from previous shifted by m elements.
For m=0 the result is A002260,
for m=1 the result is A002024,
for m=2 the result is A128076,
for m=3 the result is A131914.
This   sequence is result for m=4

			The start of the sequence as table for general case:
  1...m+1...2*m+1...3*m+1...4*m+1...5*m+1...6*m+1  ...
  2...m+2...2*m+2...3*m+2...4*m+2...5*m+2...6*m+2  ...
  3...m+3...2*m+3...3*m+3...4*m+3...5*m+3...6*m+3  ...
  4...m+4...2*m+4...3*m+4...4*m+4...5*m+4...6*m+4  ...
  5...m+5...2*m+5...3*m+5...4*m+5...5*m+5...6*m+5  ...
  6...m+6...2*m+6...3*m+6...4*m+6...5*m+6...6*m+6  ...
  7...m+7...2*m+7...3*m+7...4*m+7...5*m+7...6*m+7  ...
  ...
The start of the sequence as triangle array read by rows for general case:
     1;
    m+1,    2;
  2*m+1,   m+2,   3;
  3*m+1, 2*m+2,   m+3,   4;
  4*m+1, 3*m+2, 2*m+3,   m+4,   5;
  5*m+1, 4*m+2, 3*m+3, 2*m+4,   m+5,  6;
  6*m+1, 5*m+2, 4*m+3, 3*m+4, 2*m+5, m+6, 7;
  ...
Row number r contains r numbers: (r-1)*m+1, (r-2)*m+2,...m+r-1, r.
The start of the sequence as triangle array read by rows for m=4:
  1;
  5,2;
  9,6,3;
  13,10,7,4;
  17,14,11,8,5;
  21,18,15,12,9,6;
  25,22,19,16,13,10,7;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002260, A002024, A128076, A131914, A003056.

t=int((math.sqrt(8*n-7) - 1)/ 2)
result = +4*(t+1) + 3*(t*(t+1)/2-n)

For the general case
a(n) = m*A003056 -(m-1)*A002260.
a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n),
where t=floor((-1+sqrt(8*n-7))/2).
For m = 4
a(n) = 4*A003056 -3*A002260.
a(n) = 4*(t+1)+3*(t*(t+1)/2-n),
where t=floor((-1+sqrt(8*n-7))/2).

A341700 Sum of the primes p satisfying n < p <= 2n.

Original entry on oeis.org

2, 3, 5, 12, 7, 18, 24, 24, 41, 60, 49, 72, 59, 59, 88, 119, 102, 102, 120, 120, 161, 204, 181, 228, 228, 228, 281, 281, 252, 311, 341, 341, 341, 408, 408, 479, 515, 515, 515, 594, 553, 636, 593, 593, 682, 682, 635, 635, 732, 732, 833, 936, 883, 990, 1099, 1099

Offset: 1

Chai Wah Wu, Feb 17 2021

nonn

For n >= 2, a(n) is the sum of the prime numbers appearing in the 2nd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

			a(7) = 24 = 11+13 (sum of primes larger than 7 and less than or equal to 14).

Cf. A010051, A034387, A073837, A108954, A128076.

Array[Total@ Select[Range[# + 1, 2 #], PrimeQ] &, 56] (* Michael De Vlieger, Feb 17 2021 *)

from sympy import nextprime
def A341700(n):
    s, m = 0, nextprime(n)
    while m <= 2*n:
        s += m
        m = nextprime(m)
    return s

a(n) = A034387(2*n) - A034387(n).
a(n) = A073837(n) if n is not a prime. Otherwise, a(n) = A073837(n)-n.
For n >= 2, a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} A010051(A128076(k)) * A128076(k). - Wesley Ivan Hurt, Jan 08 2022

A070544 Number of squarefree numbers s such that n < s < 2n.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 3, 5, 6, 6, 6, 8, 7, 7, 7, 9, 9, 11, 11, 13, 13, 14, 13, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 20, 22, 22, 22, 22, 24, 23, 24, 24, 26, 27, 27, 27, 29, 30, 30, 30, 32, 32, 34, 34, 36, 36, 37, 36, 38, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 44, 44, 44, 45

Offset: 1

Benoit Cloitre, May 02 2002

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

Cf. A008683, A008966, A013928, A059956, A128076.

Table[Count[Range[n+1,2n-1],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Mar 23 2019 *)

a(n) = sum(i=n+1,2*n-1,issquarefree(i));

Limit_{n -> oo} a(n)/n = 6/Pi^2 (A059956).
From Wesley Ivan Hurt, Jan 08 2022: (Start)
a(n) = Sum_{k=1..n-1} mu(2n-k)^2.
a(n) = Sum_{k=n+1..2n-1} mu(k)^2.
a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} mu(A128076(k))^2. (End)
a(n) = A013928(2*n) - A013928(n) - A008966(n). - Amiram Eldar, Apr 29 2025

cp's OEIS Frontend

A131413 Triangle read by rows: A002024 + A128076 - A000012 as infinite lower triangular matrices.

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A060715 Number of primes between n and 2n exclusive.

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A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].

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A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

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A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

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A131914 3*A002024 - 2*A051340.

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A131914 3A002024 - 2A051340.