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A152535 a(n) = n*prime(n) - Sum_{i=1..n} prime(i). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

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.

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

EXAMPLE

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)

MATHEMATICA

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

PROG

(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

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

CROSSREFS

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

Sequence in context: A287350 A294091 A032379 * A042423 A206440 A192300

Adjacent sequences:  A152532 A152533 A152534 * A152536 A152537 A152538

KEYWORD

easy,nonn

AUTHOR

Omar E. Pol, Dec 06 2008

STATUS

approved

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Last modified February 16 03:15 EST 2019. Contains 320140 sequences. (Running on oeis4.)