OFFSET
1,1
COMMENTS
This sequence contains those 4k+1 primes p for which the first half (the (p-1)/2 most significant bits) of A055094(p) is in A014486 and those 4k+3 primes q, for which the whole A055094(q) is in A014486.
Are the 2nd, 5th and 8th primes (5,13,37) only terms of this sequence that are of the form 4k+1? [Searched up to a(211)=7927 by AK.]
No other such terms up to 19997. - Michel Marcus, Sep 21 2022
LINKS
Michel Marcus, Table of n, a(n) for n = 1..430
Antti Karttunen, Illustration of Legendre's candelabras
Eric Weisstein's World of Mathematics, Legendre Symbol
Wikipedia, Legendre symbol
MAPLE
with(numtheory); # For ithprime and legendre.
A080114v2 := proc(upto_n) local j, a, p, i, s; a := []; for i from 2 to upto_n do p := ithprime(i); s := 0; for j from 1 to (p-1)/2 do s := s + legendre(j, p); if(s < 0) then break; fi; od; if(s >= 0) then a := [op(a), p]; fi; od; RETURN(a); end;
MATHEMATICA
s[p_, u_] := Sum[JacobiSymbol[j, p], {j, 1, u}]; Select[Prime[Range[2, 200] ], (p = #; AllTrue[Range[(p - 1)/2], s[p, #] >= 0 &]) &] (* Jean-François Alcover, Mar 04 2016 *)
PROG
(Sage)
def A080114_list(n) :
a = []
for i in (2..n) :
s = 0
p = nth_prime(i)
for j in (1..(p-1)/2) :
s += legendre_symbol(j, p)
if s < 0 : break
if s >= 0 : a.append(p)
return a
A080114_list(200) # Peter Luschny, Aug 08 2012
(PARI) isok(p) = if (isprime(p) && (p>2), for (u=1, (p-1)/2, if (sum(j=1, u, kronecker(j, p)) < 0, return(0)); ); return(1); ); \\ Michel Marcus, Sep 20 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 11 2003
STATUS
approved