A279185 -id:A279185 - cp's OEIS frontend

A279186 Maximal entry in n-th row of A279185.

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

A279187 Maximal entry in row c of A279185, where c = n-th composite number A002808(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 4, 2, 6, 2, 1, 1, 4, 1, 2, 2, 6, 2, 1, 2, 4, 2, 10, 1, 6, 4, 1, 2, 6, 4, 2, 6, 3, 1, 4, 2, 1, 2, 4, 1, 10, 2, 2, 6, 4, 6, 4, 2, 1, 18, 4, 2, 1, 6, 3, 4, 2, 2, 10, 4, 11, 6, 1, 6, 4, 4, 1, 2, 2, 12, 6, 4, 6, 2, 6, 10, 3, 2

Offset: 1

N. J. A. Sloane, Dec 14 2016

nonn

There are really two sequences that should be included if missing: the maximal entry in row c, and the LCM of the entries in row c.

a(n) and the LCM variant A256608(A002808(n)) are equal at least up to n<=1100. - R. J. Mathar, Dec 15 2016

R. J. Mathar, Table of n, a(n) for n = 1..1000
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A002808, A256608 (lcm of entries in row n), A279185, A279186 (max entry in row n).

A279187 := proc(n)
    A279186(A002808(n)) ;
end proc :
seq(A279187(n),n=1..180) ; # 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]]; Composite[n_] := FixedPoint[n + PrimePi[#] + 1&, n + PrimePi[n] + 1];
a[n_] := a[n] = With[{c = Composite[n]}, Table[T[c, k], {k, 0, c-1}] // Max ];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 86}] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

A279188 Maximal entry in row c of triangle in A279185, where c = prime(n)^2 = A001248(n).

Original entry on oeis.org

1, 2, 4, 6, 20, 12, 8, 18, 110, 84, 20, 36, 20, 42, 253, 156, 812, 60, 330, 420, 18, 156, 820, 110, 48, 100, 408, 2756, 36, 84, 42, 780, 136, 1518, 1332, 60, 156, 162, 6806, 1204, 1958, 180, 3420, 96, 588, 990, 420, 1332, 3164, 684, 812, 2856, 24, 100

Offset: 1

N. J. A. Sloane, Dec 14 2016

nonn

Needs to be checked (there are really two sequences that should be included: the maximal entry in row c, and the LCM of the entries in row c).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A001248, A279185.

A279188 := proc(n)
    A279186(ithprime(n)^2) ;
end proc :
seq(A279188(n),n=1..80) ; # 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_] := a[n] = With[{c = Prime[n]^2}, Table[T[c, k], {k, 0, c-1}] // Max];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 54}] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

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)).

A256608 Longest eventual period of a^(2^k) mod n for all a.

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

Offset: 1

Ivan Neretin, Apr 04 2015

nonn

a(n) is a divisor of phi(phi(n)) (A010554).

			In other words, eventual period of {0..n-1} under the map x -> x^2 mod n.
For example, with n=10 the said map acts as follows. Read down the columns: the column headed 2 for example means that (repeatedly squaring mod 10), 2 goes to 4 goes to 16 = 6 (mod 10) goes to 36 = 6 mod 10 --- and has reached a fixed point.
0 1 2 3 4 5 6 7 8 9
0 1 4 9 6 5 6 9 4 1
0 1 6 1 6 5 6 1 6 1
0 1 6 1 6 5 6 1 6 1
and thus every number reaches a fixed point. This means the eventual common period is 1, hence a(10)=1.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A001146, A002322, A002326, A007733, A010554, A037178.
First differs from A256607 at n=43.
LCM of entries in row n of A279185.

a[n_] := With[{lambda = CarmichaelLambda[n]}, MultiplicativeOrder[2, lambda / (2^IntegerExponent[lambda, 2])]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 28 2016 *)

rpsi(n) = lcm(znstar(n)[2]); \\ A002322
pb(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ A007733
a(n) = pb(rpsi(n)); \\ Michel Marcus, Jan 28 2016

a(n) = A007733(A002322(n)).

a(prime(n)) = A037178(n). - Michel Marcus, Jan 27 2016

Name changed by Jianing Song, Feb 02 2025

A279189 Primes p such that L(p^2) = (p-1)*L(p), where L(i) = A279186(i).

Original entry on oeis.org

2, 3, 5, 29, 179, 293, 317, 467, 509, 659, 797, 1427, 1949, 2213, 2339, 2579, 2909, 3677, 4157, 4229, 4253, 4349, 5309, 5573, 5693, 5843, 5939, 6173, 6269, 6653, 6899, 6947, 7043, 7517, 7589, 8387, 8573, 8819, 9059, 9533, 10067, 10163, 10259, 10589, 11069, 11549, 11939, 13763, 14627, 15443

Offset: 1

N. J. A. Sloane, Dec 14 2016

nonn

Also, union of {2} and the primes p from A001122 such that gcd(p-1,A007733(p-1)) = 1. - Max Alekseyev, Feb 02 2024

Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Excluding a(1)=2, forms a subsequence of A001122.

Cf. A279185, A279186, A279187, A279188.

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]];
L[n_] := L[n] = Table[T[n, k], {k, 0, n - 1}] // Max;
For[p = 2, p < 1000, p = NextPrime[p], If[L[p^2] == (p-1) L[p], Print[p]]] (* Jean-François Alcover, Oct 07 2018, after Robert Israel in A279186 *)

a(8)-a(11) from Jean-François Alcover, Oct 07 2018

Terms a(12) onward from Max Alekseyev, Feb 02 2024

A343722 a(n) is the number of starting residues r modulo n from which repeated iterations of the mapping r -> r^2 mod n never reach a fixed point.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 8, 0, 8, 8, 0, 0, 0, 8, 16, 0, 12, 16, 20, 0, 16, 16, 16, 16, 24, 0, 28, 0, 24, 0, 20, 16, 32, 32, 24, 0, 32, 24, 40, 32, 20, 40, 44, 0, 40, 32, 0, 32, 48, 32, 40, 32, 48, 48, 56, 0, 56, 56, 48, 0, 40, 48, 64, 0, 60, 40, 68, 32

Offset: 1

Jon E. Schoenfield, Apr 27 2021

nonn

a(n) = 0 iff n is a term of A003401, that is, A000010(n) is a power of 2.

			For every n >= 1, the residue r = 0 is a fixed point under the mapping r -> r^2 mod n, since we have 0 -> 0^2 mod n = 0. Also, for every n >= 2, the residue r = 1 is a fixed point, since we have 1 -> 1^2 mod n = 1.
For n=1, the only residue mod n is 0 (a fixed point), so a(1) = 0.
For n=2, the only residues are 0 and 1 (each a fixed point), so a(2) = 0.
For n=3, the only residue other than 0 and 1 is 2; 2 -> 2^2 mod 3 = 4 mod 3 = 1, a fixed point, so a(3) = 0.
For n=4, we have 0 -> 0, 1 -> 1, 2 -> 2^2 mod 4 = 4 mod 4 = 0, and 3 -> 3^2 mod 4 = 9 mod 4 = 1, each trajectory ending at a fixed point, so a(4) = 0.
For n=5, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 1 -> 1
  3 -> 4 -> 1 -> 1
  4 -> 1 -> 1
(each ending at a fixed point), so a(5) = 0.
For n=6, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 4
  3 -> 3
  4 -> 4
  5 -> 1 -> 1
(each ending at a fixed point), so a(6) = 0.
For n=7, however, we have
  0 -> 0
  1 -> 1
  2 -> 4 -> 2 -> ...       (a loop)
  3 -> 2 -> 4 -> 2 -> ...  (a loop)
  4 -> 2 -> 4 -> ...       (a loop)
  5 -> 4 -> 2 -> 4 -> ...  (a loop)
  6 -> 1 -> 1
so 4 of the 7 trajectories never reach a fixed point, so a(7)=4.

Michel Marcus, Table of n, a(n) for n = 1..2000

Cf. A003401, A004169, A279185, A343720, A343721.

pos(list, r) = forstep (k=#list, 1, -1, if (list[k] == r, return (#list - k + 1)););
isok(r, n) = {my(list = List()); listput(list, r); for (k=1, oo, r = lift(Mod(r, n)^2); my(i = pos(list, r)); if (i==1, return (1)); if (i>1, return(0)); listput(list, r); );} \\ reaches a fixed point
a(n) = sum(r=0, n-1, 1 - isok(r, n)); \\ Michel Marcus, May 02 2021

a(n) is the number of terms of n-th row of A279185 that are greater than 1. - Pontus von Brömssen, Apr 27 2021

a(n) + A343721(n) = n. - Michel Marcus, May 02 2021

A279190 Primes p such that L(p^2) = (p-1)*L(p)/2, where L(i) = A279186(i).

Original entry on oeis.org

7, 11, 13, 17, 23, 47, 59, 67, 71, 83, 103, 107, 131, 139, 167, 173, 191, 227, 239, 263, 269, 347

Offset: 1

N. J. A. Sloane, Dec 14 2016

Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A279185-A279189.

A279191 Primes p such that L(p^2) = (p-1)*L(p)/4, where L(i) = A279186(i).

Original entry on oeis.org

53, 61, 97, 113, 149, 193, 349, 389, 461, 769, 773, 857, 941

Offset: 1

N. J. A. Sloane, Dec 14 2016

Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A279185, A279186, A279187, A279188, A279189, A279190, A279192.

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]];
L[n_] := L[n] = Table[T[n, k], {k, 0, n - 1}] // Max;
For[p = 2, p < 1000, p = NextPrime[p], If[L[p^2] == (p-1) L[p]/4, Print[p]]] (* Jean-François Alcover, Oct 07 2018, after Robert Israel in A279186 *)

a(8)-a(13) from Jean-François Alcover, Oct 07 2018

A279192 Primes p such that L(p^2) = (p-1)*L(p)/6, where L(i) = A279186(i).

Original entry on oeis.org

19, 31, 37, 43, 79, 199, 211, 223, 229, 277, 283, 367, 439, 463, 499, 523, 547, 619, 643, 692, 829, 859, 877, 907, 967, 997

Offset: 1

N. J. A. Sloane, Dec 14 2016

Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016.

Cf. A279185-A279191.

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]];
L[n_] := L[n] = Table[T[n, k], {k, 0, n - 1}] // Max;
For[p = 2, p < 1000, p = NextPrime[p], If[L[p^2] == (p-1) L[p]/6, Print[p]]] (* Jean-François Alcover, Oct 07 2018, after Robert Israel in A279186 *)

More terms from Jean-François Alcover, Oct 07 2018

cp's OEIS Frontend

A279186 Maximal entry in n-th row of A279185.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A279187 Maximal entry in row c of A279185, where c = n-th composite number A002808(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A279188 Maximal entry in row c of triangle in A279185, where c = prime(n)^2 = A001248(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A256608 Longest eventual period of a^(2^k) mod n for all a.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A279189 Primes p such that L(p^2) = (p-1)*L(p), where L(i) = A279186(i).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A343722 a(n) is the number of starting residues r modulo n from which repeated iterations of the mapping r -> r^2 mod n never reach a fixed point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A279190 Primes p such that L(p^2) = (p-1)*L(p)/2, where L(i) = A279186(i).

Views