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Difference between the sum of next prime(n) natural numbers and the sum of next n primes.
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%I #12 May 23 2015 15:53:31

%S 1,4,9,10,54,71,191,236,446,1025,1310,2259,3245,3820,5048,7321,10060,

%T 11473,15328,18358,20381,25672,30222,36561,46367,53031,58108,65444,

%U 70971,78391,104184,116542,133095,142728,169931,181324,203429,226622

%N Difference between the sum of next prime(n) natural numbers and the sum of next n primes.

%C Group the natural numbers with prime(n) elements in each group. (1,2),(3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17),... The sum corresponding the groups is 3,12,40,98,... Group the prime numbers such that the n-th group contains n primes. (2),(3,5),(7,11,13),(17,19,23,29),... The sum corresponding the groups is 2,8,31,88,... The required difference is 1,4,9,10,...

%C The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - _Jeremy Gardiner_, Sep 10 2004

%F a(n) = ((A061802(n-1) + 1)*A000040(n))/2 - A007468(n). - _Gionata Neri_, May 17 2015

%K nonn

%O 1,2

%A _Amarnath Murthy_, Apr 17 2003

%E More terms from _Ray Chandler_, May 13 2003