A108989 -id:A108989 - cp's OEIS frontend

A158248 Composite numbers with primitive root 10.

49, 289, 343, 361, 529, 841, 2209, 2401, 3481, 3721, 4913, 6859, 9409, 11881, 12167, 12769, 16807, 17161, 22201, 24389, 27889, 32041, 32761, 37249, 49729, 52441, 54289, 66049, 69169, 72361, 83521, 97969

Offset: 1

Robert Hutchins, Mar 15 2009

Previous name was: Numbers m whose reciprocal generates a repeating decimal fraction with period phi(m) and m/2 < phi(m) < m-1.
All terms are proper powers of full reptend primes (A001913).
This sequence does not contain every proper power of every term in A001913, for example, A001913 has 487 as its 26th term, but since 10 is not a primitive root of 487^2, 487^2 is not a term of this sequence. - Robert Hutchins, Oct 14 2021
A shorter description appears to be "Composite numbers with primitive root 10". - Arkadiusz Wesolowski, Jul 04 2012 (The two definitions certainly produce the same terms up through 83521. - N. J. A. Sloane, Jul 05 2012)

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A007732, A001913, A046145, A046146.
Subsequence of A244623.
Subsequence of A167797.
Cf. A108989 (for base 2), A346316 (for base 6).

select(n -> not isprime(n) and numtheory:-primroot(9,n) = 10,[$2..10000]);
# N. J. A. Sloane, Jul 05 2012

Select[Range[10^5], GCD[10, #] == 1 && #/2 < MultiplicativeOrder[10, #] < # - 1 &] (* Ray Chandler, Oct 17 2012 *)

More terms from Robert Hutchins, Mar 21 2009
Entry revised by N. J. A. Sloane, Jul 05 2012
New name (using comment by Arkadiusz Wesolowski) from Joerg Arndt, Nov 22 2021

A219030 Powers of odd primes (exponent > 1) for which 2 is not a primitive root.

Original entry on oeis.org

49, 289, 343, 529, 961, 1681, 1849, 2209, 2401, 4913, 5041, 5329, 6241, 7921, 9409, 10609, 11881, 12167, 12769, 16129, 16807, 18769, 22801, 24649, 27889, 29791, 36481, 37249, 39601, 49729, 52441, 54289, 57121, 58081, 63001, 66049, 68921, 69169, 73441

Offset: 1

V. Raman, Nov 10 2012

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A167791, A216848, A108989.

for(n=3,100000,if(n%2==1&&isprime(n)==0&&znorder(Mod(2,n))!=eulerphi(n)&&matsize(factor(n))[1]==1,print1(n",")))

list(lim)=my(v=List(),L=log(lim+.5));forprime(p=3,sqrtint(lim\1), for(e=2,L\log(p), if(znorder(Mod(2,p^e))Charles R Greathouse IV, Nov 12 2012

A346316 Composite numbers with primitive root 6.

Original entry on oeis.org

121, 169, 289, 1331, 1681, 2197, 3481, 3721, 4913, 6241, 6889, 7921, 10609, 11449, 11881, 12769, 14641, 16129, 17161, 18769, 22801, 24649, 28561, 32041, 39601, 49729, 51529, 52441, 54289, 63001, 66049, 68921, 73441, 76729, 83521, 120409, 134689, 139129, 157609

Offset: 1

Robert Hutchins, Jul 13 2021

nonn

An alternative description: Numbers k such that 1/k in base 6 generates a repeating fraction with period phi(n) and n/2 < phi(n) < n-1.

For example, in base 6, 1/121 has repeat length 110 = phi(121) which is > 121/2 but less than 121-1.

Robert Hutchins, PrimRoot.c

Subsequence of A244623.
Subsequence of A167794.
Cf. A108989 (for base 2),  A158248 (for base 10).
Cf. A157502.

isA033948 := proc(n)
    if n in {1,2,4} then
        true;
    elif type(n,'odd') and nops(numtheory[factorset](n)) = 1 then
        true;
    elif type(n,'even') and type(n/2,'odd') and nops(numtheory[factorset](n/2)) = 1 then
        true;
    else
        false;
    end if;
end proc:
isA167794 := proc(n)
    if not isA033948(n) or n = 1 then
        false;
    elif numtheory[order](6,n) = numtheory[phi](n) then
        true;
    else
        false;
    end if;
end proc:
A346316 := proc(n)
    option remember;
    local a;
    if n = 1 then
        121;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and isA167794(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A346316(n),n=1..20) ; # R. J. Mathar, Sep 15 2021

Select[Range[160000], CompositeQ[#] && PrimitiveRoot[#, 6] == 6 &] (* Amiram Eldar, Jul 13 2021 *)

isok(m) = (m>1) && !isprime(m) && (gcd(m, 6)==1) && (znorder(Mod(6, m))==eulerphi(m)); \\ Michel Marcus, Aug 12 2021

A167794 INTERSECT A002808.

cp's OEIS Frontend

A158248 Composite numbers with primitive root 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A219030 Powers of odd primes (exponent > 1) for which 2 is not a primitive root.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

A346316 Composite numbers with primitive root 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula