A319287 Primes of the form x^2 + y^2 where x contains only the decimal digits 0 through 6 (not 7, 8, or 9).

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557

Offset

1,1

Comments

Pratt shows that this sequence is infinite, and gives an asymptotic formula for a weighting function of a particular subset (see Theorem 1.2). This holds for any choice of at most 3 digits; Pratt comments that by imposing extra conditions on the digits chosen, a more complicated sieve argument can be used to derive a similar result.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Kyle Pratt, Primes from sums of two squares and missing digits, Proceedings of the London Mathematical Society 3:120 (2020), pp. 770-830. arXiv:1806.02699 [math.NT]

Example

113 = 7^2 + 8^2 is prime but not in this sequence because both 7 and 8 contain a digit from {7, 8, 9}.
557 = 14^2 + 19^2 is in this sequence because 557 is prime and 14 contains no digit from {7, 8, 9}.

Prog

(PARI) rd(n)=my(v=digits(n)); for(i=1, #v, if(v[i]>6, for(j=i, #v, v[j]=6); return(fromdigits(v, 7)))); fromdigits(v, 7)
list(lim)=my(v=List(), s=sqrtint(lim\=1), s7=rd(s), x2, p); for(w=1, s7, x2=fromdigits(digits(w, 7))^2; forstep(y=(x2%2)+1, sqrtint(lim-L2), 2, if(isprime(p=x2+y^2), listput(v, p)))); Set(v)

Crossrefs

Subsequence of A002313 and hence of A002144 and hence of A000040.

Cf. A007093.

Sequence in context: A351535 A002144 A280084 * A192592 A357218 A349900

Adjacent sequences: A319284 A319285 A319286 * A319288 A319289 A319290

Keyword

nonn,base

Author

Charles R Greathouse IV, Oct 10 2018

Status

approved