A111163 Triangular numbers that are sums of two consecutive primes.

36, 78, 120, 210, 276, 300, 630, 946, 990, 1770, 1830, 2556, 2850, 3240, 3570, 4278, 4950, 5460, 8256, 9870, 10878, 11026, 12090, 12720, 20100, 20910, 23436, 26796, 31626, 34980, 41616, 43660, 46056, 55278, 56616, 57630, 59340, 66066, 73920

Offset: 1

Author

Joseph L. Pe, Oct 20 2005

Keywords

• nonn

Comments

Intersection of A000217 and A001043. - Michel Marcus, May 18 2015

Examples

			36 = 8(8+1)/2 = 17 + 19. Therefore 36 belongs to the sequence.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..472 from K. D. Bajpai, n = 473..1000 from T. D. Noe)

Crossrefs

Cf. A000217 (triangular numbers), A001043 (sums of consecutive primes).

Programs

Maple

P:=proc()local i,a,b,c;c:=1;for i from 1 to 1200000 do;
a:=ithprime(i)+ithprime(i+1);b:=(-1+(sqrt(8*a+1)))/2;
if  b=floor(b) then lprint(c,a);c:=c+1;fi; od;end:P();
# K. D. Bajpai, Jun 30 2013

Mathematica

Select[Table[Prime[n] + Prime[n + 1], {n, 4500}], IntegerQ[Sqrt[1 + 8# ]] &] (* Ray Chandler, Oct 22 2005 *)
t = {}; n = 0; While[Length[t] < 100, n++; s = Prime[n] + Prime[n + 1]; If[TriangularQ[s], AppendTo[t, s]]]; t (* T. D. Noe, Jun 30 2013 *)

PARI

{p=2; ct=0; while(ct<66, q=nextprime(p+1); s=p+q; if( issquare(8*s+1), print1(s, ", "); ct++); p=q)} \\ Klaus Brockhaus, Oct 22 2005

Python

from sympy import nextprime as np
from sympy import prevprime as pp
n=1
t=n*(n+1)//2
while t>0:
    if t%2==0 and t>3:
        i=t//2
        p=pp(i); q=np(p)
        if p+q==t :
            print(t)
    n=n+1
    t=n*(n+1)//2 # Abhiram R Devesh, May 17 2015

Python

from sympy import prevprime,nextprime,isprime
A111163_list = [n*(n+1)//2 for n in range(3,10**4) if not isprime(n*(n+1)//4) and prevprime(n*(n+1)//4)+nextprime(n*(n+1)//4) == n*(n+1)//2] # Chai Wah Wu, Feb 11 2018

Extensions

Extended by Ray Chandler and Klaus Brockhaus, Oct 22 2005