A141042 -id:A141042 - cp's OEIS frontend

A162775 a(n) = A141042(n+1)/2.

2, 3, 8, 5, 12, 7, 16, 27, 10, 33, 24, 13, 28, 45, 48, 17, 54, 38, 20, 63, 44, 69, 96, 50, 26, 54, 28, 58, 210, 62, 96, 33, 170, 35, 108, 111, 76, 117, 120, 41, 210, 43, 88, 45, 276, 282, 96, 49, 100, 153, 52, 265, 162, 165, 168, 57, 174, 118, 60

Offset: 1

Omar E. Pol, Jul 30 2009

Cf. A000040, A000720, A141042, A152535.

More terms from Sean A. Irvine, Jun 22 2011

A033286 a(n) = n * prime(n).

Original entry on oeis.org

2, 6, 15, 28, 55, 78, 119, 152, 207, 290, 341, 444, 533, 602, 705, 848, 1003, 1098, 1273, 1420, 1533, 1738, 1909, 2136, 2425, 2626, 2781, 2996, 3161, 3390, 3937, 4192, 4521, 4726, 5215, 5436, 5809, 6194, 6513, 6920, 7339, 7602, 8213, 8492, 8865, 9154, 9917

Offset: 1

N. J. A. Sloane

Does an n exist such that n*prime(n)/(n+prime(n)) is an integer? - Ctibor O. Zizka, Mar 04 2008. The answer to Zizka's question is easily seen to be No: such an integer k would be positive and less than prime(n), but then k*(n + prime(n)) = prime(n)*n would be impossible. - Robert Israel, Apr 20 2015
Sums of rows of the triangle in A005145. - Reinhard Zumkeller, Aug 05 2009
Complement of A171520(n). - Jaroslav Krizek, Dec 13 2009
Partial sums of A090942. - Omar E. Pol, Apr 20 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 1.
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.

Cf. A000040, A007504, A014689, A090942, A124012, A141042, A152535.

Cf. A005145 (primes repeated), A171520 (complement), A076146 (iterated).

a033286 n = a000040 n * n  -- Reinhard Zumkeller, Jul 24 2013

[ n*NthPrime(n): n in [1..47] ]; // Klaus Brockhaus, Sep 09 2009

A033286 := proc(n) n*ithprime(n) ; end proc:
seq(A033286(n),n=1..20) ; # R. J. Mathar, Mar 21 2011

Table[Prime[n]*n, {n, 38}] (* Alonso del Arte *)

ithprime(i)*i $ i = 1..47 // Zerinvary Lajos, Feb 26 2007

a(n)=n*prime(n) \\ Charles R Greathouse IV, Jul 01 2013

a(n) = n * A000040(n) = n * A008578(n+1) = n * A158611(n+2). - Jaroslav Krizek, Aug 31 2009
a(n) = A007504(n) + A152535(n). - Omar E. Pol, Aug 09 2012
Sum_{n>=1} 1/a(n) = A124012. - Amiram Eldar, Oct 15 2020

Correction for change of offset in A158611 and A008578 in Aug 2009 from Jaroslav Krizek, Jan 27 2010

A152535 a(n) = n*prime(n) - Sum_{i=1..n} prime(i).

Original entry on oeis.org

0, 1, 5, 11, 27, 37, 61, 75, 107, 161, 181, 247, 295, 321, 377, 467, 563, 597, 705, 781, 821, 947, 1035, 1173, 1365, 1465, 1517, 1625, 1681, 1797, 2217, 2341, 2533, 2599, 2939, 3009, 3225, 3447, 3599, 3833, 4073, 4155, 4575, 4661

Offset: 1

Omar E. Pol, Dec 06 2008

a(n) is also the area under the curve of the function pi(x) from 0 to prime(n). - Omar E. Pol, Nov 13 2013

			From _Omar E. Pol_, Apr 27 2015: (Start)
For n = 5 the 5th prime is 11 and the sum of first five primes is 2 + 3 + 5 + 7 + 11 = 28, so a(5) = 5*11 - 28 = 27.
Illustration of a(5) = 27:
Consider a diagram in the first quadrant of the square grid in which the number of cells in the n-th horizontal bar is equal to the n-th prime, as shown below:
.      _ _ _ _ _ _ _ _ _ _ _
. 11  |_ _ _ _ _ _ _ _ _ _ _|
.  7  |_ _ _ _ _ _ _|* * * *
.  5  |_ _ _ _ _|* * * * * *
.  3  |_ _ _|* * * * * * * *
.  2  |_ _|* * * * * * * * *
.
a(5) is also the area (or the number of cells, or the number of *'s) under the bar's structure of prime numbers: a(5) = 1 + 4 + 6 + 16 = 27.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Christian Axler, On a sequence involving the prime numbers, arXiv:1504.04467 [math.NT], 2015 and J. Int. Seq. 18 (2015) # 15.7.6.
Christian Axler, Improving the Estimates for a Sequence Involving Prime Numbers, arXiv:1706.04049 [math.NT], 2017.

Cf. A000040, A000720, A001223, A006093, A007504, A033286, A046992, A090942, A117495, A141042 (first differences), A189892, A230849, A230850, A247380.

nn = 100; p = Prime[Range[nn]]; Range[nn] p - Accumulate[p] (* T. D. Noe, May 02 2011 *)

vector(80, n, n*prime(n) - sum(k=1, n, prime(k))) \\ Michel Marcus, Apr 20 2015

from sympy import prime, primerange
def A152535(n): return (n-1)*(p:=prime(n))-sum(primerange(p)) # Chai Wah Wu, Jan 01 2024

[n*nth_prime(n) - sum(nth_prime(j) for j in range(1,n+1)) for n in range(1,45)] # Danny Rorabaugh, Apr 18 2015

a(n) = A033286(n) - A007504(n). - Omar E. Pol, Aug 09 2012
a(n) = A046992(A006093(n)). - Omar E. Pol, Apr 21 2015
a(n+1) = Sum_{k=A000124(n-1)..A000217(n)} A204890(k). - Benedict W. J. Irwin, May 23 2016
a(n) = Sum_{k=1..n-1} k*A001223(k). - François Huppé, Mar 16 2022

A230850 A054541 and A000012 interleaved.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 4, 1, 6, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 6, 1, 4, 1, 6, 1, 8, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 14, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 6, 1, 6, 1, 4, 1, 6, 1, 6, 1, 2, 1, 10, 1, 2, 1

Offset: 1

Omar E. Pol, Oct 31 2013

nonn

a(n) is also the length of the n-th edge of a staircase which represents the function pi(x) on the first quadrant of the square grid, see A000720.
a(2n-1) is the length of the n-th horizontal edge in the staircase.
a(2n) is the length of the n-th vertical edge in the staircase.
For another version see A230849.

			Illustration of initial terms, n = 1..22:
.
1                                                              _ _|
1                                                  _ _ _ _ _ _|
1                                          _ _ _ _|
1                                      _ _|
1                              _ _ _ _|
1                          _ _|
1                  _ _ _ _|
1              _ _|
1          _ _|
1        _|
1    _ _|
.
.      2 1   2   2       4   2       4   2       4           6   2
.
Drawing vertical line segments below the staircase (as shown below) we have that the number of cells in the vertical bars gives 0 together A000720.
Drawing horizontal line segments above the staircase we have that the number of cells in the k-th horizontal bar is A000040(k).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
31  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
29  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
23  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
19  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
17  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
13  |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
11  |_ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
7   |_ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
5   |_ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
3   |_ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    0 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000012, A000040, A000720, A001223, A007504, A046992, A054541, A141042, A152535, A182986, A230849.

Riffle[Join[{2},Differences[Prime[Range[100]]]],1] (* Paolo Xausa, Oct 31 2023 *)

A230850(n) = if(1==n,2,if((n%2),prime((n+1)/2)-prime(((n+1)/2)-1),1)); \\ Antti Karttunen, Dec 23 2018

a(1) = 2; for n > 1, a(n) = A230849(n). - Antti Karttunen, Dec 23 2018

A090942 n-th arithmetic mean = prime(n).

Original entry on oeis.org

2, 4, 9, 13, 27, 23, 41, 33, 55, 83, 51, 103, 89, 69, 103, 143, 155, 95, 175, 147, 113, 205, 171, 227, 289, 201, 155, 215, 165, 229, 547, 255, 329, 205, 489, 221, 373, 385, 319, 407, 419, 263, 611, 279, 373, 289, 763, 787, 419, 327, 433, 545, 345, 781, 581, 593

Offset: 1

Amarnath Murthy, Dec 29 2003

nonn

Partial sums give A033286. - Omar E. Pol, Apr 20 2015

In other words, this is the unique sequence such that for all n >= 1, (1/n) * Sum_{k=1..n} a(k) = prime(n). - Antti Karttunen, Apr 30 2015

			From _Omar E. Pol_, Apr 20 2015: (Start)
Illustration of initial terms:
Consider a diagram in the first quadrant of the square grid in which the length of the n-th horizontal line segment is equal to the n-th prime, as shown below:
.   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  51|
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _          83|   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _      55|           |   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  33|       |           |   |
.  |_ _ _ _ _ _ _ _ _ _ _ _ _      41|   |       |           |   |
.  |_ _ _ _ _ _ _ _ _ _ _  23|       |   |       |           |   |
.  |_ _ _ _ _ _ _      27|   |       |   |       |           |   |
.  |_ _ _ _ _  13|       |   |       |   |       |           |   |
.  |_ _ _   9|   |       |   |       |   |       |           |   |
.  |_ _ 4|   |   |       |   |       |   |       |           |   |
.  |_2_|_|_ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _ _ _ _ _|_ _|
.
a(n) is also the area (or the number of cells) in the n-th region of the diagram. For example: a(4) = 7 + 6 = 13.
(End)

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A007504, A033286, A033287, A141042, A152535.

[n le 1 select 2 else n*NthPrime(n) - (n-1)*NthPrime(n-1): n in [1..60]]; // G. C. Greubel, Feb 04 2019

Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* Michael De Vlieger, Apr 20 2015 *)

vector(60, n, if(n==1,2, n*prime(n) -(n-1)*prime(n-1))) \\ G. C. Greubel, Feb 04 2019

[2] + [n*nth_prime(n) - (n-1)*nth_prime(n-1) for n in (2..60)] # G. C. Greubel, Feb 04 2019

a(n) = n*prime(n) - (n-1)*prime(n-1).
a(n) = A033287(n-2), n>1. - R. J. Mathar, Sep 08 2008
a(n) = A000040(n) + A141042(n-1), n >=2. - Omar E. Pol, Apr 20 2015

Corrected and extended by Ray Chandler, Dec 31 2003

A230849 A075526 and A000012 interleaved.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 4, 1, 6, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 6, 1, 4, 1, 6, 1, 8, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 14, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 6, 1, 6, 1, 4, 1, 6, 1, 6, 1, 2, 1, 10, 1, 2, 1

Offset: 1

Omar E. Pol, Nov 01 2013

nonn

a(n) is also the length of the n-th edge of a staircase which represents the function pi(x) on the first quadrant of the square grid, see A000720.
a(2n-1) is the length of the n-th horizontal edge in the staircase.
a(2n) is the length of the n-th vertical edge in the staircase.
For another version see A230850.

			Illustration of initial terms, n = 1..22:
.
1                                                            _ _|
1                                                _ _ _ _ _ _|
1                                        _ _ _ _|
1                                    _ _|
1                            _ _ _ _|
1                        _ _|
1                _ _ _ _|
1            _ _|
1        _ _|
1      _|
1    _|
.
.    1 1   2   2       4   2       4   2       4           6   2
.
Drawing vertical line segments below the staircase (as shown below) we have that the number of cells in the vertical bars gives A000720.
Drawing horizontal line segments above the staircase we have that the number of cells in the k-th horizontal bar is A006093(k).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
30  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
22  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
18  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
16  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
12  |_ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
10  |_ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
6   |_ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
4   |_ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
1   |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000012, A000040, A000720, A001223, A006093, A007504, A046992, A075526, A141042, A152535, A230850.

Riffle[Join[{1},Differences[Prime[Range[100]]]],1] (* Paolo Xausa, Oct 31 2023 *)

A230849(n) = if((n%2)&&(n>1),prime((n+1)/2)-prime(((n+1)/2)-1),1); \\ Antti Karttunen, Dec 23 2018

A230980 Number of primes <= n, starting at n=0.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Nov 02 2013

nonn

Essentially identical to A000720, except that sequence, being an arithmetical sequence, starts at n = 1. - N. J. A. Sloane, Jun 21 2017
Also, on the first quadrant of the square grid, consider a diagram in which the number of cells in the horizontal bar of the k-th row is equal to the k-th prime, see example. The total length of the boundary segments between the structure formed by the first k horizontal bars and the structure formed by the vertical bars, from [0, 0], is equal to A014688(k). a(n) is the number of cells in the vertical bar of the n-th column.
Note that in a similar diagram for A000720 the lengths of the horizontal bars give A006093 (primes minus 1) not A000040 (the prime numbers) because A000720 has only one zero, not two.
Also, the number of distinct prime factors of the factorial number n!. - Torlach Rush, Jan 17 2014
The lengths of the boundary horizontal segments between the structure formed by the horizontal bars and the structure formed by the vertical bars of the diagram gives A054541. The zig-zag path formed by the boundary segments is in A230850. - Omar E. Pol, Jun 22 2017

			Illustration of initial terms:
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
31   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
29   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
23   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
19   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
17   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
13   |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
11   |_ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
7    |_ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
5    |_ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
3    |_ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2    |_ _|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
n:    0 1 2 3 4 5 6 7 8 9...
a(n): 0 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A000040, A000720, A001223, A006093, A007504, A014688, A054541, A141042, A182986, A230850.

Partial sums of A010051.

[0] cat [#PrimesUpTo(n-1): n in [2..200] ]; // Vincenzo Librandi, Jun 22 2017

Array[PrimePi@# &, 90, 0] (* Robert G. Wilson v, Jun 21 2017 *)
Accumulate[Table[If[PrimeQ[n],1,0],{n,0,100}]] (* Harvey P. Dale, Mar 16 2023 *)

a(n)=primepi(n) \\ Charles R Greathouse IV, Jun 23 2017

Offset and definition changed by N. J. A. Sloane, Jun 21 2017

A147965 a(n) = n + 1 - A001223(n) = n - A046933(n). In words, a(n) is the difference between n+1 and the n-th gap between primes.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 6, 5, 4, 9, 6, 9, 12, 11, 10, 11, 16, 13, 16, 19, 16, 19, 18, 17, 22, 25, 24, 27, 26, 17, 28, 27, 32, 25, 34, 31, 32, 35, 34, 35, 40, 33, 42, 41, 44, 35, 36, 45, 48, 47, 46, 51, 44, 49, 50, 51, 56, 53, 56, 59

Offset: 1

Omar E. Pol, Nov 17 2008

Cf. A000040, A001223, A046933, A141042.

Module[{nn=70,prg},prg=Differences[Prime[Range[nn]]];#[[2]]-#[[1]]&/@ Thread[{prg,Range[nn-1]}]+1] (* Harvey P. Dale, Nov 21 2021 *)

Definition corrected by N. J. A. Sloane, Nov 21 2021, at the suggestion of Harvey P. Dale.

A147966 a(n) = n+(A001223(n)-1) = n+A046933(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 9, 8, 11, 14, 11, 16, 15, 14, 17, 20, 21, 18, 23, 22, 21, 26, 25, 28, 31, 28, 27, 30, 29, 32, 43, 34, 37, 34, 43, 36, 41, 42, 41, 44, 45, 42, 51, 44, 47, 46, 57, 58, 51, 50, 53, 56, 53, 62, 59, 60, 61, 58, 63, 62, 61, 70, 75, 66, 65, 68, 79, 72, 77

Offset: 1

Omar E. Pol, Nov 17 2008

			a(32) = A001223(32)-1+n = 6-1+32 = A046933(32)+n = 5+n = 37.

Cf. A001223, A046933, A141042, A147965.

Definition clarified by N. J. A. Sloane, Nov 22 2021 at the suggestion of Harvey P. Dale.

a(32)=37 corrected by Georg Fischer, Oct 10 2024

A147967 a(n) = n(A001223(n)-1) = nA046933(n).

Original entry on oeis.org

0, 2, 3, 12, 5, 18, 7, 24, 45, 10, 55, 36, 13, 42, 75, 80, 17, 90, 57, 20, 105, 66, 115, 168, 75, 26, 81, 28, 87, 390, 93, 160, 33, 306, 35, 180, 185, 114, 195, 200, 41, 378, 43, 132, 45, 506, 517, 144, 49, 150, 255, 52, 477

Offset: 1

Omar E. Pol, Nov 17 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001223 ("gap" with its usual meaning), A046933, A141042, A147965, A147966.

Module[{nn=60,gps},gps=Differences[Prime[Range[nn+1]]]-1;Times@@@ Thread[ {Range[nn],gps}]] (* Harvey P. Dale, Nov 07 2020 *)

Definition clarified by N. J. A. Sloane, Nov 22 2021 at the suggestion of Harvey P. Dale.

cp's OEIS Frontend

A162775 a(n) = A141042(n+1)/2.

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A033286 a(n) = n * prime(n).

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A152535 a(n) = n*prime(n) - Sum_{i=1..n} prime(i).

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A230850 A054541 and A000012 interleaved.

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A090942 n-th arithmetic mean = prime(n).

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A230849 A075526 and A000012 interleaved.

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A230980 Number of primes <= n, starting at n=0.

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A147967 a(n) = n(A001223(n)-1) = nA046933(n).