A329269 Integers k such that 8*k + 1 is a prime or a square.

0, 1, 2, 3, 5, 6, 9, 10, 11, 12, 14, 15, 17, 21, 24, 28, 29, 30, 32, 35, 36, 39, 42, 44, 45, 50, 51, 54, 55, 56, 57, 65, 66, 71, 72, 74, 75, 77, 78, 80, 84, 91, 95, 96, 101, 105, 107, 110, 116, 117, 119, 120, 122, 126, 129, 131, 136, 137, 141, 144, 149, 150

Offset: 1

Author

Frank Ellermann, Feb 23 2020

Keywords

• nonn
• easy

Comments

All odd squares have the form 8*n + 1.

Examples

			8*0 + 1 =  1 = 1^2, so 0 is a term;
8*1 + 1 =  9 = 3^2, so 1 is a term;
8*2 + 1 = 17 = prime(7), so 2 is a term;
8*3 + 1 = 25 = 5^2, so 3 is a term;
8*4 + 1 = 33 is neither prime nor square, so 4 is not a term;
8*5 + 1 = 41 = prime(13), so 5 is a term.
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, theorem 14 and ch. 4.5

Crossrefs

Union of the triangular numbers A000217 and A005123.

Cf. A000040, A016754 (odd squares).

Programs

Maple

q:= k-> (t-> isprime(t) or issqr(t))(8*k+1):
select(q, [$0..200])[];  # Alois P. Heinz, Feb 25 2020

Mathematica

Select[Range[0, 150], PrimeQ[(m = 8*# + 1)] || IntegerQ @ Sqrt[m] &] (* Amiram Eldar, Feb 29 2020 *)

PARI

isok(k) = my(x=8*k+1); isprime(x) || issquare(x); \\ Michel Marcus, Feb 27 2020

Rexx

S = 0 ;  U = 1 ;  P = 1
do N = 1 while length( S ) < 256
   C = 8 * N + 1
   do I = U by 2
      K = I * I      ;  if K > C then  leave I
      U = I          ;  if K < C then  iterate I
      S = S || ',' N ;  iterate N
   end I
   do I = P
      K = PRIME( I ) ;  if K > C then  leave I
      P = I          ;  if K < C then  iterate I
      S = S || ',' N ;  iterate N
   end I
end N
say S ;  return S