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

5, 15, 36, 67, 112, 169, 240, 323, 424, 539, 662, 803, 964, 1133, 1312, 1523, 1746, 1987, 2246, 2519, 2808, 3119, 3436, 3787, 4154, 4529, 4920, 5337, 5770, 6219, 6682, 7173, 7672, 8203, 8760, 9323, 9912, 10517, 11140, 11783, 12450, 13135, 13836

Offset: 1

Author

Zak Seidov, May 04 2005

Keywords

• nonn

Comments

Terms are squares at only(?) three values of n = 3, 6, 4072: corresponding terms are 6^2, 13^2, and 15735^2.

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.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A045943 (triangular matchstick numbers: 3*n*(n+1)/2).

Cf. A045943.

Cf. A166619, A166620. - Zak Seidov, Oct 18 2009

Programs

Magma

[ &+[ NthPrime(n+i): i in [0..n] ]: n in [1..50] ];  // Bruno Berselli, Jul 08 2011

Mathematica

a[n_]:=Plus@@Prime[Range[n, 2n]]
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 *)
Table[Total[Prime[Range[n,2n]]],{n,50}] (* Harvey P. Dale, Jun 10 2014 *)

PARI

a(n)=my(s=0);forprime(p=prime(n),prime(2*n),s+=p);s \\ Charles R Greathouse IV, Jul 08 2011

Formula

a(n) = p(n) + p(n+1) + ... + p(2n-1) + p(2n), where p(k)=k-th prime.

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