A300101 -id:A300101 - cp's OEIS frontend

A306413 a(n) is the multiplicative order of 2 modulo A001567(n).

10, 40, 28, 24, 18, 36, 28, 11, 56, 36, 60, 28, 36, 16, 230, 15, 14, 660, 36, 52, 80, 198, 30, 252, 72, 200, 60, 58, 20, 42, 22, 45, 48, 28, 96, 70, 40, 48, 460, 180, 60, 3432, 88, 72, 102, 112, 168, 44, 264, 60, 192, 21, 144, 156, 30, 153, 28, 180, 100, 22, 1012, 36, 58, 48, 60, 28, 612, 120, 60, 166, 1008, 52, 532, 148, 9840

Offset: 1

Jianing Song, Feb 13 2019

nonn

By definition, A001567 lists the odd composite numbers k such that ord(2,k) divides k - 1, where ord(2,k) is the multiplicative order of 2 modulo k. This sequence lists the values for ord(2,k) when k runs through A001567.

			A001567(1) = 341, 341 divides 2^10 - 1, 341 = 34*10 + 1.
A001567(2) = 561, 561 divides 2^40 - 1, 561 = 14*40 + 1.
A001567(3) = 645, 645 divides 2^28 - 1, 645 = 23*28 + 1.
A001567(4) = 1105, 1105 divides 2^24 - 1, 1105 = 46*24 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001567, A002326, A300101, A306413.

MultiplicativeOrder[2, #] & /@ Select[Range[1, 10^5, 2], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 &] (* Amiram Eldar, Jun 29 2019 *)

forstep(n=3, 1e5, 2, my(m=znorder(Mod(2,n))); if((n-1)%m==0 && !isprime(n), print1(m, ", ")))

a(n) = A002326((A001567(n)-1)/2).

a(n) = (A001567(n) - 1) / A300101(n). - Jianing Song, Dec 12 2021

A350083 a(n) = (A006935(n) - 1) / ord(2,A006935(n)/2), where ord(k,m) is the multiplicative order of k modulo m.

Original entry on oeis.org

1, 617, 1305, 9339, 225, 5297, 6985, 1549, 174233, 46549, 93701, 66879, 431087, 593887, 1288921, 446275, 43685, 1205, 3361, 2577225, 1313, 430739, 177301, 8541, 13067, 474525, 561301, 84725, 158873, 725725, 3851, 14019, 128861, 1090301, 2529, 430667, 541673

Offset: 1

Jianing Song, Dec 12 2021

nonn

List of (2*k-1) / ord(2,k) where k ranges over the odd numbers such that 2^(2*k-1) == 1 (mod k).

			A006935(2) = 161038, so a(2) = (161038 - 1) / ord(2,161038/2) = 617.
A006935(3) = 215326, so a(3) = (215326 - 1) / ord(2,215326/2) = 1305.

Amiram Eldar, Table of n, a(n) for n = 1..1319

Cf. A006935, A347906, A350084, A300101, A174590.

list(lim) = my(v=[],d); forstep(k=1, lim, 2, if((2*k-1)%(d=znorder(Mod(2,k)))==0, v=concat(v,(2*k-1)/d))); v \\ gives a(n) for A347906(n) <= lim

a(n) = (2*A347906(n) - 1) / ord(2,A347906(n)) = (A006935(n) - 1) / A350084(n).

A367319 Base-2 Fermat pseudoprimes k such that (k-1)/ord(2, k) > (m-1)/ord(2, m) for all base-2 Fermat pseudoprimes m < k, where ord(2, k) is the multiplicative order of 2 modulo k.

Original entry on oeis.org

341, 1105, 1387, 2047, 4369, 4681, 5461, 13981, 15709, 35333, 42799, 60787, 126217, 158369, 215265, 256999, 266305, 486737, 617093, 1082401, 1398101, 2113665, 2304167, 4025905, 4188889, 4670029, 6236473, 6242685, 8388607, 13757653, 16843009, 17895697, 22369621

Offset: 1

Amiram Eldar, Nov 14 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..182 (terms below 2^64)
Amiram Eldar, Table of n, a(n), (a(n)-1)/ord(2, a(n)) for n = 1..182

Subsequence of A001567.

Cf. A002326, A014664, A226216, A300101, A367320.

pspQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; seq[kmax_] := Module[{s = {}, r, rm = 0}, Do[If[pspQ[k], r = (k - 1)/MultiplicativeOrder[2, k]; If[r > rm, rm = r; AppendTo[s, k]]], {k, 1, kmax}]; s]; seq[10^6]

ispsp(n) = n > 1 && n % 2 && Mod(2, n)^(n-1) == 1 && !isprime(n);
lista(kmax) = {my(r, rm = 0); for(k = 1, kmax, if(ispsp(k), r = (k-1)/znorder(Mod(2, k)); if(r > rm, rm = r; print1(k, ", "))));}

cp's OEIS Frontend

A306413 a(n) is the multiplicative order of 2 modulo A001567(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350083 a(n) = (A006935(n) - 1) / ord(2,A006935(n)/2), where ord(k,m) is the multiplicative order of k modulo m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A367319 Base-2 Fermat pseudoprimes k such that (k-1)/ord(2, k) > (m-1)/ord(2, m) for all base-2 Fermat pseudoprimes m < k, where ord(2, k) is the multiplicative order of 2 modulo k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI