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.

Previous Showing 11-20 of 29 results. Next

A240100 Numbers with primitive root -17.

Original entry on oeis.org

2, 4, 5, 10, 19, 25, 37, 38, 41, 43, 47, 50, 59, 61, 67, 74, 82, 83, 86, 94, 97, 103, 113, 118, 122, 125, 127, 134, 151, 166, 173, 179, 191, 193, 194, 197, 206, 226, 233, 239, 250, 251, 254, 263, 269, 271, 277, 302, 313, 317, 331, 346, 358, 359, 361, 382
Offset: 1

Views

Author

Vincenzo Librandi, Apr 01 2014

Keywords

Links

Crossrefs

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15), A240101 (r=17).
Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15).

Programs

  • Mathematica
    pr = -17; Select[Range[2, 400], MultiplicativeOrder[pr, #] == EulerPhi[#] &]

A279186 Maximal entry in n-th row of A279185.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 2, 1, 1, 1, 2, 6, 1, 2, 4, 10, 1, 4, 2, 6, 2, 3, 1, 4, 1, 4, 1, 2, 2, 6, 6, 2, 1, 4, 2, 6, 4, 2, 10, 11, 1, 6, 4, 1, 2, 12, 6, 4, 2, 6, 3, 28, 1, 4, 4, 2, 1, 2, 4, 10, 1, 10, 2, 12, 2, 6, 6, 4, 6, 4, 2, 12, 1, 18, 4, 20, 2, 1, 6, 3, 4
Offset: 1

Views

Author

N. J. A. Sloane, Dec 14 2016

Keywords

Comments

See A256608 for LCM of entries in row n.
From Robert Israel, Dec 15 2016: (Start)
If m and k are coprime then a(m*k) = lcm(a(m), a(k)).
If n is in A061345 and r = A053575(n) is in A167791, then a(n) = A000010(r). (End)

Links

Crossrefs

Cf. A000010, A053575, A063145, A167791, A279185.
Start is same as A256607 and A256608. However, all three are different.

Programs

  • Maple
    A279186 := proc(n)
    local a,k ;
    a := 1 ;
    for k from 0 to n-1 do
        a := max(a,A279185(k,n)) ;
    end do:
    a ;
end proc : # R. J. Mathar, Dec 15 2016
  • Mathematica
    T[n_, k_] := Module[{g, y, r}, If[k == 0, Return[1]]; y = n; g = GCD[k, y]; While[g > 1, y = y/g; g = GCD[k, y]]; If[y == 1, Return[1]]; r = MultiplicativeOrder[k, y]; r = r/2^IntegerExponent[r, 2]; If[r == 1, Return[1]]; MultiplicativeOrder[2, r]];
a[n_] := Table[T[n, k], {k, 0, n - 1}] // Max;
Array[a, 90] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)
  • PARI
    { A279186(n) = my(r=lcm(znstar(n)[2])); znorder(Mod(2,r>>valuation(r,2))); } \\ Max Alekseyev, Feb 02 2024

Formula

a(n) = A007733(A002322(n)). - Max Alekseyev, Feb 02 2024

A179383 a(n) = 2*k(n)-1 where k(n) is the sequence of positions of records in A179382.

Original entry on oeis.org

1, 5, 9, 11, 13, 19, 25, 29, 37, 53, 59, 61, 67, 83, 101, 107, 121, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757
Offset: 1

Views

Author

Vladimir Shevelev, Jul 12 2010

Keywords

Comments

Records in A179382(k(n)) = 1, 2, 3, 5, 6, 9, 10, 14, 18, 26, 29, ....
are located at k(n) = 1, 3, 5, 6, 7, 10, 13, 15, 19, 27, 30, 31,..
The current sequence is a simple transformation of this k(n) sequence.
Question: Are there any terms in the sequence with two or more distinct prime divisors?
Some very plausible conjectures: 1) The sequence consists of primes and squares of primes; 2) The set of squares is finite; 3) A prime p>=5 is in the sequence iff it has primitive root 2 (A001122) ; 4) There exists l such that, for n>l, A179383(n) =A139099(n+l) . [From Vladimir Shevelev , Jul 14 2010]

Links

Crossrefs

Cf. A139099, A167791, A002326, A179382

Extensions

Definition rephrased and sequence extended by R. J. Mathar, Jul 13 2010
I made a change to Conjecture 4). - Vladimir Shevelev, Jul 18 2010

A270096 Smallest m such that 2^m == 2^n (mod n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 4, 5, 2, 9, 4, 1, 2, 1, 5, 3, 2, 11, 6, 1, 2, 3, 4, 1, 6, 1, 4, 9, 2, 1, 4, 7, 10, 3, 4, 1, 18, 15, 5, 3, 2, 1, 4, 1, 2, 3, 6, 5, 6, 1, 4, 3, 10, 1, 6, 1, 2, 15, 4, 17, 6, 1, 4
Offset: 1

Views

Author

Thomas Ordowski, Mar 11 2016

Keywords

Comments

a(n) = 1 iff n is a prime or a pseudoprime (odd or even) to base 2.
We have a(n) <= n - phi(n) and a(n) <= phi(n), so a(n) <= n/2.
From Robert Israel, Mar 11 2016: (Start)
If n is in A167791, then a(n) = A068494(n).
If n is odd, a(n) = n mod A002326((n-1)/2).
a(n) >= A007814(n).
a(p^k) = p^(k-1) for all k >= 1 and all odd primes p not in A001220.
Conjecture: a(n) <= n/3 for all n > 8. (End)

Links

Crossrefs

Cf. A000010, A001220, A002326, A007814, A051953, A068494, A167791.
Cf. A276976 (a generalization on all integer bases).

Programs

  • Maple
    f:= proc(n) local d,b,t, m,c;
  d:= padic:-ordp(n,2);
  b:= n/2^d;
  t:= 2 &^ n mod n;
  m:= numtheory:-mlog(t,2,b,c);
  if m < d then m:= m + c*ceil((d-m)/c) fi;
  m
end proc:
f(1):= 0:
map(f, [$1..1000]; # Robert Israel, Mar 11 2016
  • Mathematica
    Table[k = 0; While[PowerMod[2, n, n] != PowerMod[2, k, n], k++]; k, {n, 120}] (* Michael De Vlieger, Mar 15 2016 *)
  • PARI
    a(n) = {my(m = 0); while (Mod(2, n)^m != 2^n, m++); m; } \\ Altug Alkan, Sep 23 2016

Formula

a(n) < n/2 for n > 4.
a(2^k) = k for all k >= 0.
a(2*p) = 2 for all primes p.

Extensions

More terms from Michel Marcus, Mar 11 2016

A240102 Numbers with primitive root -18.

Original entry on oeis.org

5, 7, 23, 29, 31, 37, 47, 53, 61, 71, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 223, 239, 263, 269, 271, 277, 293, 317, 349, 359, 367, 383, 389, 397, 421, 461, 479, 503, 509, 529, 541, 557, 607, 613, 647, 653, 661, 677, 701, 719, 733
Offset: 1

Views

Author

Vincenzo Librandi, Apr 01 2014

Keywords

Links

Crossrefs

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15), A240101 (r=17), A240103 (r=18).
Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15), A240100 (r=-17).

Programs

  • Mathematica
    pr = -18; Select[Range[2, 800], MultiplicativeOrder[pr, #] == EulerPhi[#] &]
  • PARI
    is(n)=if(gcd(n,6)>1, return(0)); my(p=eulerphi(n)); znorder(Mod(-18,n),p)==p \\ Charles R Greathouse IV, Nov 26 2014

A240103 Numbers with primitive root 18.

Original entry on oeis.org

5, 11, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 121, 139, 149, 157, 163, 173, 179, 181, 197, 227, 251, 269, 277, 283, 293, 317, 347, 349, 379, 389, 397, 419, 421, 461, 467, 491, 509, 523, 541, 547, 557, 563, 571, 587, 613, 619, 653, 659, 661, 677
Offset: 1

Views

Author

Vincenzo Librandi, Apr 01 2014

Keywords

Links

Crossrefs

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15), A240101 (r=17).
Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15), A240100 (r=-17), A240102 (r=-18).

Programs

  • Mathematica
    pr = 18; Select[Range[2, 800], MultiplicativeOrder[pr, #] == EulerPhi[#] &]
  • PARI
    is(n)=if(gcd(n, 6)>1, return(0)); my(p=eulerphi(n)); znorder(Mod(18, n), p)==p \\ Charles R Greathouse IV, Nov 26 2014

A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 10, 2, 1, 1, 1, 6, 3, 2, 1, 9, 2, 3, 3, 2, 2, 6, 1, 13, 9, 1, 1, 10, 5, 1, 3, 2, 8, 3, 2, 2, 1, 1, 10, 3, 8, 7, 9, 2, 2, 3, 1, 2, 26, 1, 3, 9, 4, 2, 9, 4, 1, 6, 1, 18, 9, 1, 7, 3, 2, 1, 3, 2, 5, 10, 1, 10, 6, 38, 3, 3, 4, 1, 41, 2
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 15 2014

Keywords

Comments

a(n) is the number of integers k in range 1 .. A016754(n-1)-1 such that A000225(k) is an integral multiple of A016754(n-1). - Antti Karttunen, Nov 15 2014
Conjecture: the positions of 1's, a(k)=1, are exactly given by the 2k-1 which are elements of A167791. - Antti Karttunen, Nov 15 2014
From Charlie Neder, Oct 18 2018: (Start)
It would appear that, if 2k-1 is in A167791, then so is (2k-1)^2, and so a(k) = 1 would follow by definition.
Conjecture: Let B be the first value such that (2k-1)^2 divides 2^B - 1. Then either 2k-1 divides B, or 2k-1 is a Wieferich prime (A001220). (End)

Examples

			a(2) = 1 because (2^6 - 1)/(2*2 - 1)^2 = 7 is an integer and 6 < 9.
a(3) = 1 because (2^20 - 1)/(2*3 - 1)^2 = 41943 is an integer and 20 < 25.
a(3) = 2 because (2^21 - 1)/(2*4 - 1)^2 = 42799 is an integer and 21 < 49; and also (2^42 - 1)/(2*4 - 1)^2 = 89756051247 is an integer and 42 < 49.

Links

Crossrefs

A246703 gives the positions of records.
Cf. A000225, A002326, A016754, A049094, A237043.

Programs

  • Maple
    A246702 := proc(n)
    local a,klim,k ;
    a := 0 ;
    klim := (2*n-1)^2 ;
    for k from 1 to klim-1 do
        if modp(2^k-1,klim) = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A246702(n),n=1..80) ; # R. J. Mathar, Nov 15 2014
  • Mathematica
    A246702[n_] := Module[{a, klim, k}, a = 0; klim = (2*n-1)^2; For[k = 1, k <= klim-1, k++, If[Mod[2^k-1, klim] == 0, a = a+1]]; a];
Table[A246702[n], {n, 1, 84}] (* Jean-François Alcover, Oct 04 2017, translated from R. J. Mathar's Maple code *)
  • PARI
    a(n)=my(t=(2*n-1)^2,m=Mod(1,t)); sum(k=1,t-1,m*=2;m==1) \\ Charles R Greathouse IV, Nov 16 2014
  • PARI
    a246702(n) = my(m=(2*n-1)^2); (m-1)\znorder(Mod(2,m)); \\ Max Alekseyev, Oct 11 2023
  • Scheme
    (define (A246702 n) (let ((u (A016754 (- n 1)))) (let loop ((k (- u 1)) (s 0)) (cond ((zero? k) s) ((zero? (modulo (A000225 k) u)) (loop (- k 1) (+ s 1))) (else (loop (- k 1) s)))))) ;; Antti Karttunen, Nov 15 2014

Formula

a(n) = floor( 4*n*(n-1) / A002326(2*n*(n-1)) ). - Max Alekseyev, Oct 11 2023

Extensions

Corrected by R. J. Mathar, Nov 15 2014

A240101 Numbers with primitive root 17.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 14, 22, 23, 25, 31, 37, 41, 46, 49, 50, 61, 62, 74, 82, 97, 98, 107, 113, 121, 122, 125, 131, 139, 167, 173, 193, 194, 197, 211, 214, 226, 227, 233, 242, 250, 262, 269, 277, 278, 283, 311, 313, 317, 334, 343, 346, 347, 367, 379, 386
Offset: 1

Views

Author

Vincenzo Librandi, Apr 01 2014

Keywords

Links

Crossrefs

Cf. numbers with positive primitive root r: A167791 (r=2), A167792 (r=3), A167793 (r=5), A167794 (r=6), A167795 (r=7), A167796 (r=8), A167797 (r=10), A240028 (r=11), A240030 (r=12), A240032 (r=13), A240094 (r=14), A240096 (r=15).
Cf. numbers with negative primitive root r: A167798 (r=-2), A167799 (r=-3), A167800 (r=-4), A167801 (r=-5), A167802 (r=-6), A167803 (r=-7), A167804 (r=-8), A167805 (r=-9), A167806 (r=-10), A240029 (r=-11), A240031 (r=-12), A240093 (r=-13), A240095 (r=-14), A240097 (r=-15), A240100 (r=-17).

Programs

  • Mathematica
    pr = 17; Select[Range[2, 400], MultiplicativeOrder[pr, #] == EulerPhi[#] &]

A084550 Numbers n that have a primitive root less than n that are nonnegative perfect eleventh powers.

Original entry on oeis.org

2053, 2083, 2099, 2131, 2141, 2187, 2197, 2213, 2221, 2237, 2243, 2269, 2293, 2309, 2339, 2357, 2371, 2389, 2437, 2459, 2467, 2477, 2539, 2549, 2557, 2579, 2621, 2659, 2677, 2683, 2693, 2699, 2741, 2789, 2797, 2803, 2809, 2819, 2837, 2843, 2851
Offset: 1

Views

Author

Hauke Worpel (hw1(AT)email.com), May 30 2003

Keywords

Comments

Includes n if n is in A167791 with n > 2048 and phi(n) not divisible by 11.

Links

Crossrefs

Cf. A167791.

Programs

  • Maple
    filter:= proc(n) local i,t,s;
  s:= numtheory:-phi(n);
  for i from 2 do
    t:= i^11;
    if t >= n then return false fi;
    if numtheory:-order(t,n) = s then return true fi;
  od
end proc:
select(filter, [$2..10000]); # Robert Israel, Mar 02 2021
  • Mathematica
    filterQ[n_] := Module[{i, t, s}, s = EulerPhi[n]; For[i = 2, True, i++, t = i^11; If[t >= n, Return[False]]; If[MultiplicativeOrder[t, n] == s, Return[True]]]];
Select[Range[2, 10000], filterQ] (* Jean-François Alcover, Feb 02 2023, after Robert Israel *)

A108989 Composite numbers k with primitive root 2; i.e., the order of 2 modulo k is phi(k).

Original entry on oeis.org

9, 25, 27, 81, 121, 125, 169, 243, 361, 625, 729, 841, 1331, 1369, 2187, 2197, 2809, 3125, 3481, 3721, 4489, 6561, 6859, 6889, 10201, 11449, 14641, 15625, 17161, 19321, 19683, 22201, 24389, 26569, 28561, 29929, 32041, 32761, 38809, 44521, 50653
Offset: 1

Views

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Jul 28 2005

Keywords

Comments

There exist no even numbers with primitive root 2. All entries are odd. They are all the powers of odd primes. - V. Raman, Nov 20 2012

Examples

			Modulo 9: 2^1 == 2, 2^2 == 4, 2^3 == 8, 2^4 == 7, 2^5 == 5, 2^6 == 1 and phi(9) == 6.

Links

Crossrefs

Intersection of A002808 and A167791.
Cf. A000010, A001122, A002326, A216838, A216848, A219030.

Programs

  • GAP
    for i in [2..100000] do if not IsPrime(i) then if IsPrimitiveRootMod(2,i) then Display(i); fi; fi; od;
  • Mathematica
    nn=51000; Select[Complement[Range[2, nn], Prime[Range[PrimePi[nn]]]], PrimitiveRoot[#] == 2&] (* Harvey P. Dale, Jul 25 2011 *)
seq[max_] := Module[{ps = Select[Range[2, Floor[Sqrt[max]]], PrimeQ], s = {}}, Do[s = Join[s, Select[p^Range[2, Floor[Log[p, max]]], PrimitiveRoot[#] == 2 &]], {p, ps}]; Sort[s]]; seq[10^5] (* Amiram Eldar, Nov 10 2023 *)
  • PARI
    for(n=3,100000,if(n%2==1&&isprime(n)==0&&znorder(Mod(2,n))==eulerphi(n),print1(n","))) /* V. Raman, Nov 20 2012 */
Previous Showing 11-20 of 29 results. Next

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

Content is available under The OEIS End-User License Agreement.