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

1,1

Comments

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

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)

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A255101 A255094 A261284 * A286430 A279901 A335276

Adjacent sequences: A111160 A111161 A111162 * A111164 A111165 A111166

Keyword

nonn,changed

Author

Joseph L. Pe, Oct 20 2005

Extensions

Extended by Ray Chandler and Klaus Brockhaus, Oct 22 2005

Status

approved