A007733 -id:A007733 - cp's OEIS frontend

A068563 Numbers k such that 2^k == 4^k (mod k).

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 42, 48, 54, 60, 64, 72, 80, 84, 96, 100, 108, 120, 126, 128, 136, 144, 156, 160, 162, 168, 180, 192, 200, 216, 220, 240, 252, 256, 272, 288, 294, 300, 312, 320, 324, 336, 342, 360, 378, 384, 400, 408, 420, 432, 440

Offset: 1

Benoit Cloitre, Mar 25 2002

If k is in the sequence then 2k is also in the sequence, but the converse is not true.
Contains A124240 as a subsequence. Their difference is given by A124241. - T. D. Noe, May 30 2003
Also, integers k such that A007733(k) divides k. Also, integers k such that for every odd prime divisor p of k, A007733(p) = A002326((p-1)/2) divides k. Also, integers k such that A000265(k) divides 2^k-1. - Max Alekseyev, Aug 25 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Carmichael Function.

Cf. A000265, A002322, A002326, A007733, A124240, A124241.

Select[Range[500], PowerMod[2,#,# ] == PowerMod[4,#,# ] & ]

isok(k) = Mod(2, k)^k == Mod(4, k)^k; \\ Amiram Eldar, Apr 19 2025

Comment and Mathematica program corrected by T. D. Noe, Oct 17 2008

A204987 Least k such that n divides 2^k - 2^j for some j satisfying 1 <= j < k.

Original entry on oeis.org

2, 2, 3, 3, 5, 3, 4, 4, 7, 5, 11, 4, 13, 4, 5, 5, 9, 7, 19, 6, 7, 11, 12, 5, 21, 13, 19, 5, 29, 5, 6, 6, 11, 9, 13, 8, 37, 19, 13, 7, 21, 7, 15, 12, 13, 12, 24, 6, 22, 21, 9, 14, 53, 19, 21, 6, 19, 29, 59, 6, 61, 6, 7, 7, 13, 11, 67, 10, 23, 13, 36, 9, 10, 37, 21, 20, 31, 13, 40, 8, 55, 21, 83, 8, 9, 15, 29, 13

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

See A204892 for a discussion and guide to related sequences.

			1 divides 2^2 - 2^1, so a(1)=2;
2 divides 2^2 - 2^1, so a(2)=2;
3 divides 2^3 - 2^1, so a(3)=3;
4 divides 2^3 - 2^2, so a(4)=3;
5 divides 2^5 - 2^1, so a(5)=5.

Antti Karttunen, Table of n, a(n) for n = 1..6556

Cf. A000079, A007733, A007814, A054703, A204892, A204988.

s[n_] := s[n] = 2^n; z1 = 1000; z2 = 50;
Table[s[n], {n, 1, 30}]     (* A000079 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204985 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]     (* A204986 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]     (* A204987 *)
Table[j[n], {n, 1, z2}]     (* A204988 *)
Table[s[k[n]], {n, 1, z2}]  (* A204989 *)
Table[s[j[n]], {n, 1, z2}]  (* A140670 ? *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204991 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204992 *)
%%/2  (* A204990=(1/2)*A204991 *)

A204987etA204988(n) = { my(k=2); while(1,for(j=1,k-1,if(!(((2^k)-(2^j))%n),return([k,j]))); k++); }; \\ (Computes also A204988 at the same time) - Antti Karttunen, Nov 19 2017

a(n)={my(k=valuation(n,2)); max(k, 1) + znorder(Mod(2, n>>k))} \\ Andrew Howroyd, Aug 08 2018

a(n) = max(1, A007814(n)) + A007733(n). - Andrew Howroyd, Aug 08 2018

More terms from Antti Karttunen, Nov 19 2017

A007737 Period of repeating digits of 1/n in base 6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 10, 1, 12, 2, 1, 1, 16, 1, 9, 1, 2, 10, 11, 1, 5, 12, 1, 2, 14, 1, 6, 1, 10, 16, 2, 1, 4, 9, 12, 1, 40, 2, 3, 10, 1, 11, 23, 1, 14, 5, 16, 12, 26, 1, 10, 2, 9, 14, 58, 1, 60, 6, 2, 1, 12, 10, 33, 16, 11, 2, 35, 1, 36, 4, 5, 9, 10, 12, 78, 1, 1, 40, 82, 2, 16, 3, 14, 10

Offset: 1

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Not multiplicative. Smallest counterexample: a(77)=10, but a(7) = 2 and a(11) = 10. - Mitch Harris, May 16 2005.

Cf. A007733 (base 2), A007734 (3), A007735 (4), A007736 (5), A007738 (7), A007739 (8), A007740 (9), A007732 (10).

DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 6] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[6, n/Times @@ ({2, 3}^IntegerExponent[n, {2, 3}])]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)

a(n)=znorder(Mod(6, n/2^valuation(n, 2)/3^valuation(n, 3))); \\ Joerg Arndt, Dec 14 2014

More terms from David W. Wilson

A066799 Square array read by antidiagonals of eventual period of powers of k mod n; period of repeating digits of 1/n in base k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 2, 1, 4, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 1, 1, 1, 1, 4, 1, 6, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 2, 2, 3, 4, 10, 1, 1, 1, 2, 1, 2, 2, 1, 1, 6, 2, 5, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 12

Offset: 1

Henry Bottomley, Dec 20 2001

The determinant of the n X n matrix made from the northwest corner of this array is 0^(n-1). - Iain Fox, Mar 12 2018

			Rows start: 1,1,1,1,1,...; 1,1,1,1,1,...; 1,2,1,1,2,...; 1,1,2,1,1; 1,4,4,2,1,... T(3,2)=2 since the powers of 2 become 1,2,1,2,1,2,... mod 3 with period 2. T(4,2)=1 since the powers of 2 become 1,2,0,0,0,0,... mod 4 with eventual period 1.
Beginning of array:
+-----+--------------------------------------------------------------------
| n\k |  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  ...
+-----+--------------------------------------------------------------------
|  1  |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
|  2  |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
|  3  |  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1, ...
|  4  |  1,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2,  1, ...
|  5  |  1,  4,  4,  2,  1,  1,  4,  4,  2,  1,  1,  4,  4,  2,  1,  1, ...
|  6  |  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1,  2,  1,  1, ...
|  7  |  1,  3,  6,  3,  6,  2,  1,  1,  3,  6,  3,  6,  2,  1,  1,  3, ...
|  8  |  1,  1,  2,  1,  2,  1,  2,  1,  1,  1,  2,  1,  2,  1,  2,  1, ...
| ... |

Iain Fox, Antidiagonals n = 1..141 of array, flattened

Columns are A000012, A007733, A007734, A007735, A007736, A007737, A007738, A007739, A007740, A007732. A002322 is the highest value in each row and the least common multiple of each row, while the number of distinct values in each row is A066800.

t[n_, k_] := For[p = PowerMod[k, n, n]; m = n + 1, True, m++, If[PowerMod[k, m, n] == p, Return[m - n]]]; Flatten[Table[t[n - k + 1, k], {n, 1, 14}, {k, n, 1, -1}]] (* Jean-François Alcover, Jun 04 2012 *)

a(n, k) = my(p=k^n%n); for(m=n+1, +oo, if(k^m%n==p, return(m-n))) \\ Iain Fox, Mar 12 2018

T(n, k) = T(n, k-n) if k > n.
T(n, n) = T(n, n+1) = 1.
T(n, n-1) = 2.

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

N. J. A. Sloane, Dec 14 2016

nonn

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)

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A000010, A053575, A063145, A167791, A279185.

Start is same as A256607 and A256608. However, all three are different.

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

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 *)

{ A279186(n) = my(r=lcm(znstar(n)[2])); znorder(Mod(2,r>>valuation(r,2))); } \\ Max Alekseyev, Feb 02 2024

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

A336693 Period of binary representation of 1/(1+sigma(n)).

Original entry on oeis.org

1, 1, 4, 1, 3, 12, 6, 1, 3, 18, 12, 28, 4, 20, 20, 1, 18, 4, 6, 14, 10, 36, 20, 60, 1, 14, 20, 18, 5, 9, 10, 1, 21, 20, 21, 11, 12, 60, 18, 12, 14, 48, 12, 8, 39, 9, 21, 100, 28, 23, 9, 30, 20, 110, 9, 110, 54, 12, 60, 156, 6, 48, 12, 1, 8, 28, 22, 7, 48, 28, 9, 21, 20, 44, 100, 46, 48, 156, 54, 40, 60, 7, 8, 60

Offset: 1

Antti Karttunen, Jul 31 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A007733, A088580, A324294, A332459, A336691, A336692, A336694, A336695.

A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
A336693(n) = A007733(1+sigma(n));

a(n) = A007733(1+A000203(n)) = A007733(A088580(n)) = A007733(A332459(n)).

A007735 Period of base 4 representation of 1/n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 6, 3, 2, 1, 4, 3, 9, 2, 3, 5, 11, 1, 10, 6, 9, 3, 14, 2, 5, 1, 5, 4, 6, 3, 18, 9, 6, 2, 10, 3, 7, 5, 6, 11, 23, 1, 21, 10, 4, 6, 26, 9, 10, 3, 9, 14, 29, 2, 30, 5, 3, 1, 6, 5, 33, 4, 11, 6, 35, 3, 9, 18, 10, 9, 15, 6, 39, 2, 27, 10, 41, 3, 4, 7, 14, 5, 11, 6, 6, 11, 5

Offset: 1

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Cf. A007733 (base 2), A007734 (3), A007736 (5), A007737 (6), A007738 (7), A007739 (8), A007740 (9), A007732 (10).

DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 4] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[4, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)

More terms from David W. Wilson

A007740 Period of repeating digits of 1/n in base 9.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 5, 1, 3, 3, 2, 2, 8, 1, 9, 2, 3, 5, 11, 1, 10, 3, 1, 3, 14, 2, 15, 4, 5, 8, 6, 1, 9, 9, 3, 2, 4, 3, 21, 5, 2, 11, 23, 2, 21, 10, 8, 3, 26, 1, 10, 3, 9, 14, 29, 2, 5, 15, 3, 8, 6, 5, 11, 8, 11, 6, 35, 1, 6, 9, 10, 9, 15, 3, 39, 2, 1, 4, 41, 3, 8, 21, 14, 5, 44, 2, 3, 11, 15, 23

Offset: 1

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Not multiplicative. Smallest counterexample a(5) = a(16) = 2, but a(80) = 2. - David W. Wilson, Jun 09 2005

Cf. A007733 (base 2), A007734 (3), A007735 (4), A007736 (5), A007737 (6), A007738 (7), A007739 (8), A007732 (10).

DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 9] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[9, n/3^IntegerExponent[n, 3]]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)

A256607 Eventual period of 2^(2^k) mod n.

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, 3, 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, 3

Offset: 1

Ivan Neretin, Apr 04 2015

nonn

In other words, eventual period of 2 under the map x -> x^2 mod n.

a(n) is a divisor of A256608(n).

			For n=9 the map acts as follows: 2 -> 4 -> 7 -> 4 -> 7 and so on. This means the eventual period is 2, hence a(9)=2.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001146, A007733, A002326.
First differs from A256608 at n=43.
Column 2 of triangle in A279185.

a256607 = a007733 . fromIntegral . a007733
-- Reinhard Zumkeller, Apr 13 2015

a(n) = A007733(A007733(n)).

A007739 Period of repeating digits of 1/n in base 8.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 1, 2, 4, 10, 2, 4, 1, 4, 1, 8, 2, 6, 4, 2, 10, 11, 2, 20, 4, 6, 1, 28, 4, 5, 1, 10, 8, 4, 2, 12, 6, 4, 4, 20, 2, 14, 10, 4, 11, 23, 2, 7, 20, 8, 4, 52, 6, 20, 1, 6, 28, 58, 4, 20, 5, 2, 1, 4, 10, 22, 8, 22, 4, 35, 2, 3, 12, 20, 6, 10, 4, 13, 4, 18, 20, 82, 2, 8, 14, 28, 10, 11, 4, 4

Offset: 1

N. J. A. Sloane, Hal Sampson (hals(AT)easynet.com)

Cf. A007733 (base 2), A007734 (3), A007735 (4), A007736 (5), A007737 (6), A007738 (7), A007740 (9), A007732 (10).

DigitCycleLength[r_Rational, b_Integer?Positive] := MultiplicativeOrder[b, FixedPoint[ Quotient[#, GCD[#, b]] &, Denominator[r]]]; DigitCycleLength[1, b_Integer?Positive] = 1; Array[ DigitCycleLength[1/#, 8] &, 80] (* Robert G. Wilson v, Jun 10 2011 *)
a[n_] := MultiplicativeOrder[8, n/2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Aug 26 2024 *)

More terms from David W. Wilson

cp's OEIS Frontend

A068563 Numbers k such that 2^k == 4^k (mod k).

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A204987 Least k such that n divides 2^k - 2^j for some j satisfying 1 <= j < k.

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A007737 Period of repeating digits of 1/n in base 6.

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A066799 Square array read by antidiagonals of eventual period of powers of k mod n; period of repeating digits of 1/n in base k.

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A279186 Maximal entry in n-th row of A279185.

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A336693 Period of binary representation of 1/(1+sigma(n)).

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A007735 Period of base 4 representation of 1/n.

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A007740 Period of repeating digits of 1/n in base 9.