A080114 - OEIS

%I #30 Sep 21 2022 09:33:41

%S 3,5,7,11,13,23,31,37,47,59,71,79,83,103,131,151,167,191,199,239,251,

%T 263,271,311,359,383,419,431,439,479,503,563,599,607,647,659,719,743,

%U 751,839,863,887,911,919,971,983,991,1031,1039,1063,1091,1103,1151,1223

%N Odd primes p for which all sums Sum_{j=1..u} L(j/p) (with u ranging from 1 to (p-1)/2) are nonnegative, where L(j/p) is Legendre symbol of j and p.

%C 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.

%C 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.]

%C No other such terms up to 19997. - _Michel Marcus_, Sep 21 2022

%H Michel Marcus, <a href="/A080114/b080114.txt">Table of n, a(n) for n = 1..430</a>

%H Antti Karttunen, <a href="/A080120/a080120.pdf">Illustration of Legendre's candelabras</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendreSymbol.html">Legendre Symbol</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_symbol">Legendre symbol</a>

%p with(numtheory); # For ithprime and legendre.

%p A080114 := n -> ithprime(A080112(n));

%p 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;

%t 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 *)

%o (Sage)

%o def A080114_list(n) :

%o     a = []

%o     for i in (2..n) :

%o         s = 0

%o         p = nth_prime(i)

%o         for j in (1..(p-1)/2) :

%o             s += legendre_symbol(j, p)

%o             if s < 0 : break

%o         if s >= 0 : a.append(p)

%o     return a

%o A080114_list(200) # _Peter Luschny_, Aug 08 2012

%o (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

%Y Cf. A080112, A080115. These are the primes for which a "Legendre's candelabra" can be constructed, see A080120.

%Y Supersequence of A095102.

%K nonn

%O 1,1

%A _Antti Karttunen_, Feb 11 2003