This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A082749 #16 Apr 13 2025 15:06:16
%S A082749 1,4,9,10,54,71,191,236,446,1025,1310,2259,3245,3820,5048,7321,10060,
%T A082749 11473,15328,18358,20381,25672,30222,36561,46367,53031,58108,65444,
%U A082749 70971,78391,104184,116542,133095,142728,169931,181324,203429,226622
%N A082749 Difference between the sum of next prime(n) natural numbers and the sum of next n primes.
%C A082749 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 A082749 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 A082749 a(n) = ((A061802(n-1) + 1)*A000040(n))/2 - A007468(n). - _Gionata Neri_, May 17 2015
%t A082749 Module[{nn=80,trms=40,c,nat,pr},c=(nn(nn+1))/2;nat=Total/@TakeList[Range[c],Prime[Range[trms]]];pr=Total/@TakeList[Prime[Range[c]], Range[trms]]; Differences/@ Thread[{pr,nat}]]//Flatten (* _Harvey P. Dale_, Apr 13 2025 *)
%K A082749 nonn
%O A082749 1,2
%A A082749 _Amarnath Murthy_, Apr 17 2003
%E A082749 More terms from _Ray Chandler_, May 13 2003