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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A139035 Primes of the form 4*k+3 with primitive root -2. 3
7, 23, 47, 71, 79, 103, 167, 191, 199, 239, 263, 271, 311, 359, 367, 383, 463, 479, 487, 503, 599, 607, 647, 719, 743, 751, 823, 839, 863, 887, 967, 983, 991, 1031, 1039, 1063, 1087, 1151, 1223, 1231, 1279, 1303, 1319, 1367, 1439, 1447, 1487, 1511, 1543, 1559 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Original name: Primes with semiprimitive root 2.

If p is a prime, then we call r a semiprimitive root if it has order (p-1)/2 and there is no x for which a^x is congruent to -1 (mod p).  So +/- r^k, 0 <= k <= (p-3)/2 is a complete set of nonzero residues (mod p).

If r=2, then (-1/p)=-1 and, consequently, a(n)==-1(mod 4).

Besides, (2/a(n))=1. Indeed, since 2^((p-1)/2)=1 (mod p), then 2==2^((p+1)/2)=(2^((p+1)/4))^2. Therefore, (a(n))^2==1(mod 16) and thus a(n)==-1(mod 8). This yields that residues 1,2,...,2^((p-3)/2) are quadratic residues modulo a(n), while -1,-2,...,-2^((p-3)/2) are quadratic nonresidues modulo a(n). Primitive root of a(n) is greater than or equal to 3. All terms are in A115591.

REFERENCES

Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv preprint arXiv:1608.00862, 2016

LINKS

Table of n, a(n) for n=1..50.

Christian Elsholtz, Almost all primes have a multiple of small Hamming weight, arXiv:1602.05974 [math.NT], 2016. See p. 6.

V. Shevelev, On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2, arXiv:0710.1354 [math.NT], 2007.

FORMULA

Prime p is in the sequence iff p==-1(mod 8) and A002326((p-1)/2)=(p-1)/2. A sufficient condition: if p==-1 (mod 8) and (p-1)/2 is prime, then p is in the sequence (the converse statement, generally speaking, is not true).

A006694((a(n)-1)/2)=2 and A064287((a(n)-1)/2)=1.

MATHEMATICA

Reap[For[p = 3, p <= 10^4, p = NextPrime[p], rp = MultiplicativeOrder[2, p]; rm = MultiplicativeOrder[-2, p]; If[rp != p-1 && rm == p-1, Sow[p]]] ][[2, 1]] (* Jean-Fran├žois Alcover, Sep 03 2016, after Joerg Arndt *)

PROG

(PARI)

{ forprime (p=3, 10^4,

    rp = znorder(Mod(+2, p));

    rm = znorder(Mod(-2, p));

    if ( (rp!=p-1) && (rm==p-1), print1(p, ", ") );

); }

/* Joerg Arndt, Jun 03 2012 */

CROSSREFS

Cf. A006694, A002326, A064287, A001917, A001122, A133954, A115591.

Sequence in context: A000353 A097149 A185007 * A002146 A184882 A073577

Adjacent sequences:  A139032 A139033 A139034 * A139036 A139037 A139038

KEYWORD

nonn,changed

AUTHOR

Vladimir Shevelev, May 31 2008, Jun 06 2008

EXTENSIONS

New name by Joerg Arndt, Jun 03 2012

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 July 26 02:08 EDT 2017. Contains 289798 sequences.