A230849 -id:A230849 - cp's OEIS frontend

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

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

cp's OEIS Frontend

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

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