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

1,3

Comments

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

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

Links

Table of n, a(n) for n=1..62.

Example

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.

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 *)

Prog

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

Crossrefs

Union of the triangular numbers A000217 and A005123.

Cf. A000040, A016754 (odd squares).

Sequence in context: A276476 A069880 A230044 * A069861 A047449 A162923

Adjacent sequences: A329266 A329267 A329268 * A329270 A329271 A329272

Keyword

nonn,easy

Author

Frank Ellermann, Feb 23 2020

Status

approved