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%I #13 Sep 18 2018 04:15:30
%S -2,-4,-5,-6,-1,-4,3,-2,7,32,21,50,57,40,49,86,123,96,137,142,109,156,
%T 161,210,305,304,253,254,201,204,497,490,545,472,663,582,641,700,685,
%U 746,807,708,937,830,809,700,1041,1382,1347,1216
%N Difference between n times the n-th prime and twice the sum of the first n primes.
%C Rosser & Schoenfeld prove Robert Mandl's conjecture that a(n) > 0 for n > 8. - _Charles R Greathouse IV_, Jun 11 2015
%H Robert Israel, <a href="/A240568/b240568.txt">Table of n, a(n) for n = 1..10000</a>
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="https://doi.org/10.1090/S0025-5718-1975-0457373-7">Sharper bounds for the Chebyshev functions θ(x) and Ψ(x)</a>, Mathematics of Computation 29 (1975), pp. 243-269.
%H Nilotpal Kanti Sinha, <a href="http://arxiv.org/abs/1011.1667">On the asymptotic expansion of the sum of the first n primes</a>, arXiv:1011.1667 [math.NT], 2010-2015.
%F a(n) = n * A000040(n) - 2 * A007504(n).
%F a(n) ~ n^2 / 2. - _Charles R Greathouse IV_, Jun 11 2015
%e For n = 3, the third prime is 5 and the sum of the first 3 primes is 10, so a(3) = -5
%p N:= 1000; # to use the primes up to N
%p Primes:= select(isprime,[$1..N]);
%p CPrimes:= Statistics:-CumulativeSum(Primes):
%p A240568:= [seq(round(n*Primes[n] - 2*CPrimes[n]),n=1..nops(Primes))]; # _Robert Israel_, Apr 08 2014
%t a[n_] := a[n] = a[n-1] + (n-2) Prime[n] - (n-1) Prime[n-1]; a[1] = -2;
%t Array[a, 50] (* _Jean-François Alcover_, Sep 18 2018 *)
%o (PARI) a(n)=my(q=prime(n),s); forprime(p=2,q,s+=p); q*n-2*s \\ _Charles R Greathouse IV_, Jun 11 2015
%Y Cf. A000040, A007504.
%K sign
%O 1,1
%A _Robert Israel_, Apr 08 2014
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