A226069 Primes p such that p-1 is a square and p-2 is a triangular number.

2, 5, 17, 17957, 4726277, 23911673957

Offset: 1

Author

Alex Ratushnyak, May 25 2013

Keywords

• nonn

Comments

Roots of squares: a(n)-1 = b(n)*b(n), b(n) = A226070(n).

Roots of triangular numbers: a(n)-2 = c(n)*(c(n)+1)/2, c(n) = A226071(n).

Primes of the form A006452(k)^2+1. a(7) is too large to include here (see b-file). - Max Alekseyev, Jan 30 2014

Links

• Max Alekseyev, Table of n, a(n) for n = 1..12

Crossrefs

Cf. A226070-A226074.

Programs

C

#include 
#include 
#include 
#define TOP (1ULL<<34)
int main() {
  unsigned long long i, j, k, r, n=1;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (i=3; i < TOP*2; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      ++n;
      if (i<(1ULL<<32))
        for (j=i*i>>1; j>3] |= 1 << (j&7);
    }
  //printf("%llu\n", n);
  for (i=1, j=i*i+1; j < TOP*2; i++,j=i*i+1)
    if(j==2 || ((j&1) && (c[j>>4] & (1<<((j>>1) & 7)))==0)) {
      k = j-2;
      r = sqrt(k*2);
      if (r*r+r==k*2) printf("%9llu %9llu %9llu\n", r, i, j);
    }
  free(c);
  return 0;
}

Extensions

Terms a(7)-a(12) from Max Alekseyev, Jan 30 2014