A329269 - OEIS

%I #54 Sep 06 2023 13:41:12

%S 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,

%T 50,51,54,55,56,57,65,66,71,72,74,75,77,78,80,84,91,95,96,101,105,107,

%U 110,116,117,119,120,122,126,129,131,136,137,141,144,149,150

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

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

%D 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

%e 8*0 + 1 =  1 = 1^2, so 0 is a term;

%e 8*1 + 1 =  9 = 3^2, so 1 is a term;

%e 8*2 + 1 = 17 = prime(7), so 2 is a term;

%e 8*3 + 1 = 25 = 5^2, so 3 is a term;

%e 8*4 + 1 = 33 is neither prime nor square, so 4 is not a term;

%e 8*5 + 1 = 41 = prime(13), so 5 is a term.

%p q:= k-> (t-> isprime(t) or issqr(t))(8*k+1):

%p select(q, [$0..200])[];  # _Alois P. Heinz_, Feb 25 2020

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

%o (Rexx)

%o S = 0 ;  U = 1 ;  P = 1

%o do N = 1 while length( S ) < 256

%o    C = 8 * N + 1

%o    do I = U by 2

%o       K = I * I      ;  if K > C then  leave I

%o       U = I          ;  if K < C then  iterate I

%o       S = S || ',' N ;  iterate N

%o    end I

%o    do I = P

%o       K = PRIME( I ) ;  if K > C then  leave I

%o       P = I          ;  if K < C then  iterate I

%o       S = S || ',' N ;  iterate N

%o    end I

%o end N

%o say S ;  return S

%o (PARI) isok(k) = my(x=8*k+1); isprime(x) || issquare(x); \\ _Michel Marcus_, Feb 27 2020

%Y Union of the triangular numbers A000217 and A005123.

%Y Cf. A000040, A016754 (odd squares).

%K nonn,easy

%O 1,3

%A _Frank Ellermann_, Feb 23 2020