A233415 -id:A233415 - cp's OEIS frontend

A081856 Numbers k such that 2k-1 divides 2^k-1.

1, 2, 8, 128, 228, 648, 1352, 1908, 3240, 4608, 5220, 5976, 11448, 13160, 13920, 21528, 22050, 23760, 23940, 24840, 30960, 31284, 31584, 31968, 32768, 37224, 46092, 46512, 47268, 60480, 65664, 66528, 78540, 78600, 81728, 82800, 84312, 98406, 102672, 103968

Offset: 1

Benoit Cloitre, Apr 11 2003

nonn

Subsequence of odd terms is given by A233415. - Charles R Greathouse IV, Dec 04 2013

Numbers 2k-1 form a subsequence of A187787. - Max Alekseyev, Sep 04 2024

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A187787, A233415.

a:= proc(n) option remember; local k;
      if n=1 then 1 else for k from 1+a(n-1)
      while 2&^k mod(2*k-1)<>1 do od; k fi
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, May 27 2016

terms = 100; Reap[For[n=1; k=1, k <= terms, n++, If[Divisible[2^n-1, 2n-1], Print[k, " ", n]; Sow[n]; k++]]][[2, 1]] (* Jean-François Alcover, Apr 06 2017 *)
Join[{1},Select[Range[110000],PowerMod[2,#,2*#-1]==1&]] (* Harvey P. Dale, Jan 19 2019 *)

is(n)=Mod(2,2*n-1)^n==1 \\ Charles R Greathouse IV, Dec 04 2013

a(38)-a(40) from Michel Marcus, Dec 04 2013

A273614 Numbers k such that 3k - 1 divides 3^k - 1.

Original entry on oeis.org

1, 9, 12, 96, 345, 432, 852, 945, 1452, 2160, 3480, 3753, 4800, 6561, 6984, 13230, 15840, 17040, 30210, 31008, 40320, 43776, 44352, 44652, 47628, 55200, 56940, 60420, 61065, 69312, 71145, 74100, 77400, 81504, 125580, 128016, 175952, 192240, 198168

Offset: 1

Juri-Stepan Gerasimov, May 26 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)

Cf. A024023, A081856, A233415.

[n: n in [1..200000] | Modexp(3, n, 3*n-1) eq 1];

a:= proc(n) option remember; local k;
      if n=1 then 1 else for k from 1+a(n-1)
      while 3&^k mod(3*k-1)<>1 do od; k fi
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, May 27 2016

Select[Range[10^6], PowerMod[3, #, 3*# - 1] == 1 &] (* Giovanni Resta, May 27 2016 *)

is(n)=Mod(3,3*n-1)^n==1 \\ Charles R Greathouse IV, May 29 2016

A273727 Numbers k such that 3k - 1 divides 3^k - 1 and 3 does not divide k.

Original entry on oeis.org

1, 175952, 348880, 649078, 951625, 1495472, 1944320, 3140852, 3483200, 3643270, 4359040, 4703776, 6513584, 8170904, 9854200, 11005568, 11831890, 12149872, 12828200, 12910928, 18095630, 18964400, 22034516, 43599424, 62849024, 66322480, 82159588, 85181600

Offset: 1

Juri-Stepan Gerasimov, May 28 2016

nonn

Cf. A233415. Subsequence of A273614.

Select[Range[10^6], Mod[#,3] > 0 && PowerMod[3, #, 3*#-1] == 1 &] (* Giovanni Resta, May 29 2016 *)

is(n)=n%3 && Mod(3,3*n-1)^n==1 \\ Charles R Greathouse IV, May 29 2016

A273772 Least k > 1 such that n(kn-1) - 1 divides n^(k*n-1) - 1, or 0 if no such k exists.

Original entry on oeis.org

381713, 58651, 12301, 2861, 1656278791, 547, 5179643, 214, 2719331, 26627, 73651287679, 90205, 5069, 5533707, 13117, 58385, 791716066017, 5589, 21214381292, 3802401, 509437122973, 167, 1261552, 6001, 1144853, 3111, 6952504, 143573

Offset: 2

Juri-Stepan Gerasimov, May 29 2016

Is gcd(n, a(n)) = 1? - David A. Corneth, Jun 02 2016
No: gcd(15, a(15)) = 3. - Charles R Greathouse IV, Jun 04 2016
If a(30) is not 0, then it exceeds 5 * 10^12.  Terms a(31) through a(34) are 4219, 124522631, 305201, and 6475739899. - Lucas A. Brown, Feb 28 2024
1.1*10^16 < a(30) <= 123676617214883421968465. - Max Alekseyev, Sep 24 2024

Cf. A233415, A273727.

a(n)=my(k=2,n2=n^2); while(Mod(n,k*n2-n-1)^(k*n-1)!=1, k++); k \\ Charles R Greathouse IV, Jun 02 2016

a(3) corrected by Charles R Greathouse IV, Jun 02 2016
a(6)-a(17) from Charles R Greathouse IV, Jun 04 2016
a(18)-a(29) from Lucas A. Brown, Feb 28 2024

cp's OEIS Frontend

A081856 Numbers k such that 2k-1 divides 2^k-1.

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A273614 Numbers k such that 3k - 1 divides 3^k - 1.

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A273727 Numbers k such that 3k - 1 divides 3^k - 1 and 3 does not divide k.

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A273772 Least k > 1 such that n*(k*n-1) - 1 divides n^(k*n-1) - 1, or 0 if no such k exists.

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A273772 Least k > 1 such that n(kn-1) - 1 divides n^(k*n-1) - 1, or 0 if no such k exists.