login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A080114 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, which is defined to be 1 if j is a quadratic residue (mod p) and -1 if j is a quadratic non-residue (mod p). 7
3, 5, 7, 11, 13, 23, 31, 37, 47, 59, 71, 79, 83, 103, 131, 151, 167, 191, 199, 239, 251, 263, 271, 311, 359, 383, 419, 431, 439, 479, 503, 563, 599, 607, 647, 659, 719, 743, 751, 839, 863, 887, 911, 919, 971, 983, 991, 1031, 1039, 1063, 1091, 1103, 1151, 1223 (list; graph; refs; listen; history; text; internal format)
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.]

LINKS

Table of n, a(n) for n=1..54.

A. Karttunen, Illustration of Legendre's candelabras

MAPLE

with(numtheory); # For ithprime and legendre.

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

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-Fran├žois 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

CROSSREFS

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

Supersequence of A095102.

Sequence in context: A182583 A181741 A154319 * A088878 A259730 A254673

Adjacent sequences:  A080111 A080112 A080113 * A080115 A080116 A080117

KEYWORD

nonn

AUTHOR

Antti Karttunen Feb 11 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 14 00:21 EST 2017. Contains 295976 sequences.