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A152535
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a(n) = n*prime(n) - Sum_{i=1..n} prime(i).
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12
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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 |_ _|* * * * * * * * *
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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)
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MATHEMATICA
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nn = 100; p = Prime[Range[nn]]; Range[nn] p - Accumulate[p] (* T. D. Noe, May 02 2011 *)
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PROG
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(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
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CROSSREFS
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Cf. A000040, A000720, A001223, A006093, A007504, A033286, A046992, A090942, A117495, A141042 (first differences), A189892, A230849, A230850, A247380.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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