A226153 Numbers n such that triangular(n) is an average of 4 consecutive primes.

5, 10, 14, 15, 22, 34, 49, 54, 64, 66, 81, 93, 104, 116, 121, 122, 146, 154, 156, 180, 194, 221, 222, 236, 270, 299, 320, 332, 334, 337, 346, 360, 369, 371, 374, 387, 416, 417, 429, 435, 444, 472, 492, 498, 511, 520, 551, 556, 617, 622, 637, 654, 657, 670, 674, 677, 680

Offset: 1

Author

Alex Ratushnyak, May 28 2013

Keywords

• nonn

Links

• Zak Seidov, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A102655, A226151.

Programs

C

#include 
#include 
#include 
#define TOP (1ULL<<30)
int main() {
  unsigned long long i, j, p1, p2, p3, r, s;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (i=3; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0 /*&& i<(1ULL<<32)*/)
        for (j=i*i>>1; j>3] |= 1 << (j&7);
  for (p3=2, p2=3, p1=5, i=7; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      s = p3 + p2 + p1 + i;
      if (s%4==0) {
        s/=4;
        r = sqrt(s*2);
        if (r*(r+1)==s*2) printf("%llu, ", r);
      }
      p3 = p2, p2 = p1, p1 = i;
    }
  return 0;
}

Maple

A034963 := proc(n)
    add(ithprime(i), i=n..n+3) ;
end proc:
istriangular:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then return t1 ; else return -1; end if; end;
for n from 1 to 90000 do
    s := A034963(n)/4 ;
    if type(s,'integer') then
    tr := istriangular(s) ;
    if tr >= 0  then
        printf("%d, ", tr) ;
    end if;
    end if;
end do: # R. J. Mathar, Jun 06 2013

Mathematica

Module[{nn=30000,ntrs,m},ntrs=Table[{n,(n(n+1))/2},{n,nn}];m=Mean/@Partition[Prime[ Range[ nn]],4,1];Select[ntrs,MemberQ[m,#[[2]]]&]][[;;,1]] (* Harvey P. Dale, Jun 08 2023 *)
(Sqrt[8#+1]-1)/2&/@Select[Mean/@Partition[Prime[Range[25000]],4,1],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Sep 17 2024 *)