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

Author

Omar E. Pol, Dec 06 2008

Keywords

• easy
• nonn

Comments

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

Examples

			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)
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

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

Python

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

Sage

[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

Formula

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