cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Robert Hutchins

Robert Hutchins's wiki page.

Robert Hutchins has authored 5 sequences.

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

Views

Author

Robert Hutchins, Jul 13 2021

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    Select[Range[160000], CompositeQ[#] && PrimitiveRoot[#, 6] == 6 &] (* Amiram Eldar, Jul 13 2021 *)
  • PARI
    isok(m) = (m>1) && !isprime(m) && (gcd(m, 6)==1) && (znorder(Mod(6, m))==eulerphi(m)); \\ Michel Marcus, Aug 12 2021

Formula

A167794 INTERSECT A002808.

A345674 Euler totient function phi(n) - number of primitive roots modulo n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 4, 4, 2, 6, 4, 8, 4, 8, 8, 8, 4, 12, 8, 12, 6, 12, 8, 12, 8, 12, 12, 16, 8, 22, 16, 20, 8, 24, 12, 24, 12, 24, 16, 24, 12, 30, 20, 24, 12, 24, 16, 30, 12, 32, 24, 28, 12, 40, 24, 36, 16, 30, 16, 44, 22, 36, 32, 48, 20, 46, 32, 44, 24
Offset: 1

Views

Author

Robert Hutchins, Jun 22 2021

Keywords

Crossrefs

Cf. A000010, A046144.

Programs

  • Maple
    a:= proc(n) uses numtheory; `if`(n=1, 0, (p->
      p-add(`if`(order(i, n)=p, 1, 0), i=0..n-1))(phi(n)))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jun 22 2021
  • Mathematica
    a[n_] := (e = EulerPhi[n]) - If[n == 1 || IntegerQ @ PrimitiveRoot[n], EulerPhi[e], 0]; Array[a, 100] (* Amiram Eldar, Jun 23 2021 *)

Formula

a(n) = A000010(n) - A046144(n).

A158899 These are numbers n such that the reciprocal, 1/n, is a repeating fraction whose period is n/2 - 1.

Original entry on oeis.org

14, 34, 38, 46, 58, 94, 118, 122, 194, 218, 226, 262, 298, 334, 358, 362, 386, 446, 458, 466, 514, 526, 538, 626, 674, 734, 758, 766, 778, 838, 866, 922, 974, 982, 998, 1006, 1018, 1082, 1142, 1154, 1186, 1238, 1294, 1318, 1402, 1418, 1454, 1486, 1622, 1642
Offset: 1

Views

Author

Robert Hutchins, Mar 29 2009

Keywords

Comments

These numbers relate to the long period primes, those that for 1/m the period is m-1 (sequence A006883) in that by multiplying each term in the long period primes by 2, this sequence is generated.

Crossrefs

Cf. A006883, A007732, A085837, A121090.

Programs

  • PARI
    forstep(n=2, 2e3, 2, if ((setminus(Set(factor(n)[,1]), Set([2,5])) != []) && (znorder(Mod(10, n/2^valuation(n, 2)/5^valuation(n, 5))) + 1 == n/2), print1(n, ", "));); \\ Michel Marcus, Feb 24 2013

Extensions

More terms and edited by Michel Marcus, Feb 24 2013

A158248 Composite numbers with primitive root 10.

Original entry on oeis.org

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

Views

Author

Robert Hutchins, Mar 15 2009

Keywords

Comments

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)

Links

Crossrefs

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

Programs

  • Maple
    select(n -> not isprime(n) and numtheory:-primroot(9,n) = 10,[$2..10000]);
# N. J. A. Sloane, Jul 05 2012
  • Mathematica
    Select[Range[10^5], GCD[10, #] == 1 && #/2 < MultiplicativeOrder[10, #] < # - 1 &] (* Ray Chandler, Oct 17 2012 *)

Extensions

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

A174839 Lists of 4 adjacent primes such that the difference between the highest minus the lowest = 8.

Original entry on oeis.org

3, 5, 7, 11, 5, 7, 11, 13, 11, 13, 17, 19, 101, 103, 107, 109, 191, 193, 197, 199, 821, 823, 827, 829, 1481, 1483, 1487, 1489, 1871, 1873, 1877, 1879, 2081, 2083, 2087, 2089, 3251, 3253, 3257, 3259
Offset: 1

Views

Author

Robert Hutchins, Mar 30 2010

Keywords

Links

Programs

  • C
    if (primes[i+3] - primes[i] == 8) { printf("%u, %u, %u, %u, \n", primes[i] , primes[i+1], primes[i+2] , primes[i+3]); }
  • Mathematica
    Flatten[Select[Partition[Prime[Range[500]],4,1],Last[#]-First[#]==8&]] (* Harvey P. Dale, Oct 01 2013 *)

Formula

a(n) >> n log^4 n. - Charles R Greathouse IV, Dec 13 2024

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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