A226070 -id:A226070 - cp's OEIS frontend

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

Alex Ratushnyak, May 25 2013

nonn

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

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

Cf. A226070-A226074.

#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;
}

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

A226071 Numbers k such that triangular(k)+1 is a square and triangular(k)+2 is a prime.

Original entry on oeis.org

0, 2, 5, 189, 3074, 218685

Offset: 1

Alex Ratushnyak, May 25 2013

Subsequence of A006451 that contains the terms k such that triangular(k)+2 is prime.

a(7) is too large to include here (see b-file). - Max Alekseyev, Jan 30 2014

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

Corresponding primes: A226069.
Roots of corresponding squares: A226070.
Cf. A226071-A226074.

C
```
// see A226069.
```

Select[Range[0,220000],IntegerQ[Sqrt[(#(#+1))/2+1]]&&PrimeQ[(#(#+1))/2+2]&] (* Harvey P. Dale, Mar 01 2023 *)

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

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

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