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A105720 Triangular matchstick numbers in the class of prime numbers: sum of n-th and next n primes.

%I A105720 #20 Sep 08 2022 08:45:17
%S A105720 5,15,36,67,112,169,240,323,424,539,662,803,964,1133,1312,1523,1746,
%T A105720 1987,2246,2519,2808,3119,3436,3787,4154,4529,4920,5337,5770,6219,
%U A105720 6682,7173,7672,8203,8760,9323,9912,10517,11140,11783,12450,13135,13836
%N A105720 Triangular matchstick numbers in the class of prime numbers: sum of n-th and next n primes.
%C A105720 Terms are squares at only(?) three values of n = 3, 6, 4072: corresponding terms are 6^2, 13^2, and 15735^2.
%C A105720 Terms are prime at many values of n; at n = 1, 4, 16, 18, 22, 36, 40, 44, 52 they are 5, 67, 1523, 1987, 3119, 9323, 11783, 14551, 21019.
%H A105720 Harvey P. Dale, <a href="/A105720/b105720.txt">Table of n, a(n) for n = 1..1000</a>
%F A105720 a(n) = p(n) + p(n+1) + ... + p(2n-1) + p(2n), where p(k)=k-th prime.
%F A105720 a(1)=5; for n > 1, a(n) = a(n-1) - prime(n-1) + prime(2*n-1) + prime(2*n). - _Zak Seidov_, Oct 18 2009
%t A105720 a[n_]:=Plus@@Prime[Range[n, 2n]]
%t A105720 a=5;s={5};Do[a=a-Prime[n]+Prime[2n+1]+Prime[2n+2];AppendTo[s,a],{n,10^5}]; (* _Zak Seidov_, Oct 18 2009 *)
%t A105720 Table[Total[Prime[Range[n,2n]]],{n,50}] (* _Harvey P. Dale_, Jun 10 2014 *)
%o A105720 (Magma) [ &+[ NthPrime(n+i): i in [0..n] ]: n in [1..50] ];  // _Bruno Berselli_, Jul 08 2011
%o A105720 (PARI) a(n)=my(s=0);forprime(p=prime(n),prime(2*n),s+=p);s \\ _Charles R Greathouse IV_, Jul 08 2011
%Y A105720 Cf. A045943 (triangular matchstick numbers: 3*n*(n+1)/2).
%Y A105720 Cf. A045943.
%Y A105720 Cf. A166619, A166620. - _Zak Seidov_, Oct 18 2009
%K A105720 nonn
%O A105720 1,1
%A A105720 _Zak Seidov_, May 04 2005