A002326 -id:A002326 - cp's OEIS frontend

A137605 Consider the sequence: b(0) = n, and for k >= 1, b(k) = b(k-1)/2 if b(k-1) is even, otherwise b(k) = n - (b(k-1)+1)/2. Then a(n) = m, where m is the smallest index such that b(m) = 1.

0, 1, 1, 2, 2, 4, 5, 3, 3, 8, 5, 10, 9, 8, 13, 4, 4, 11, 17, 11, 9, 6, 11, 22, 20, 7, 25, 19, 8, 28, 29, 5, 5, 32, 21, 34, 8, 19, 29, 38, 26, 40, 7, 27, 10, 11, 9, 35, 23, 14, 49, 50, 11, 52, 17, 35, 13, 43, 11, 23, 54, 19, 49, 6, 6, 64, 17, 35, 33, 68, 45, 59, 13, 41, 73, 14, 23, 19, 25

Offset: 1

Yasutoshi Kohmoto, Apr 23 2008

nonn

The first occurrence of the numbers 0, 1, 2, 3, 4, ... is at indices 1, 2, 4, 8, 6, 7, 22, 26, 10, 13, 12, 18, 1366, 15, 50, 386, ... - Robert G. Wilson v, May 15 2008

			6->3->4->2->1. So a(6)=4.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000.
Wikipedia, Faro shuffle.

Cf. A002326, A003558, A137604.

f[n_] := Block[{lst = {n}, a}, While[a = lst[[ -1]]; a != 1, If[EvenQ@a, AppendTo[lst, a/2], AppendTo[lst, lst[[1]] - (a + 1)/2]]]; Length@ lst - 1]; Array[f, 79] (* Robert G. Wilson v, May 15 2008 *)

For n>2, if 2n-1 is in A014657, then a(n) = A002326(n-1)/2 - 1, otherwise a(n) = A002326(n-1) - 1. In particular, if A002326(n-1) is odd, then a(n) = A002326(n-1) - 1. - Max Alekseyev, May 21 2008, Dec 09 2017

For n>2, a(n) = A003558(n-1) - 1. - Joerg Arndt, Sep 12 2013

More terms from Robert G. Wilson v, May 15 2008

Edited by Max Alekseyev, Dec 09 2017

A217469 Multiplicative order of 5 (mod 5*n + 1).

Original entry on oeis.org

2, 5, 4, 6, 4, 3, 6, 20, 22, 16, 6, 30, 10, 5, 9, 54, 42, 12, 8, 25, 52, 36, 14, 55, 6, 65, 16, 46, 72, 75, 4, 66, 82, 18, 20, 15, 6, 19, 42, 22, 102, 35, 18, 16, 112, 30, 29, 40, 20, 25, 64, 42, 18, 27, 22, 140, 20, 96, 36, 42, 48, 155, 39, 106, 54, 165, 12, 15

Offset: 1

Arkadiusz Wesolowski, Nov 16 2012

nonn

Least m such that 5*n + 1 divides 5^m - 1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A003571, A003573.

List([1..80],n->OrderMod(5,5*n+1)); # Muniru A Asiru, Feb 26 2019

Table[MultiplicativeOrder[5, 5*n + 1], {n, 68}]

vector(80, n, znorder(Mod(5, 5*n+1))) \\ Michel Marcus, Feb 09 2015

A246717 Numbers of the form 2n - 1 such that A246702(n) = 2.

Original entry on oeis.org

7, 17, 23, 35, 41, 47, 49, 71, 77, 79, 95, 97, 103, 115, 137, 143, 167, 175, 191, 193, 199, 209, 235, 239, 245, 263, 271, 289, 295, 299, 311, 313, 319, 335, 343, 359, 367, 371, 383, 395, 401, 407, 409, 413, 415, 437, 449, 463, 475, 479, 487, 503, 515, 517, 521, 529, 535, 539, 551, 569, 575, 581, 583, 599, 607, 611, 647, 649, 667, 695, 707

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

From Antti Karttunen, Nov 15 2014: (Start)
Equally: Odd numbers n for which A246702((n+1)/2) = 2.
Primes in this sequence: 7, 17, 23, 41, 47, 71, 79, ... seem to be A115591.
A249819 gives the composite terms.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..105

Cf. A002326, A115591, A249819, A167791, A246702.

isA246717(n) = { if(!(n%2), return(0), my(u, s=0); u = n^2; for(k=1, u-1, if(!(((2^k)-1)%u), s++;if(s > 2, return(0)))); return(2==s)); }
n = 0; i = 0; while(i < 105, n++; if(isA246717(n), i++; write("b246717.txt", i, " ", n))); \\ From Antti Karttunen, Nov 15 2014
(Scheme, with Antti Karttunen's IntSeq-library)
(define A246717 (MATCHING-POS 1 1 (lambda (n) (and (odd? n) (= 2 (A246702 (/ (+ 1 n) 2)))))))

Terms corrected by Antti Karttunen, Nov 15 2014

A292267 Restricted growth sequence transform of A292268; filter combining multiplicative order of 2 mod 2n+1 and the number of trailing 1's in binary expansion of 2n+1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 12, 10, 13, 14, 15, 16, 17, 18, 12, 19, 7, 20, 21, 22, 23, 24, 25, 26, 27, 28, 7, 29, 30, 31, 32, 33, 34, 35, 36, 37, 9, 38, 39, 16, 15, 40, 41, 42, 43, 44, 7, 45, 17, 46, 13, 47, 7, 48, 49, 33, 43, 50, 51, 52, 25, 53, 54, 55, 56, 57, 13, 58, 59, 60, 61, 33, 23, 62, 63, 64, 12, 65, 66, 10, 67, 57, 68, 69, 70, 71, 17, 72, 25

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768

Cf. A002326, A007814, A292268.

Cf. A291766, A291769 for related filters.

allocatemem(2^30);
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
A007814(n) = valuation(n,2);
A292268(n) = (1/2)*(2 + ((A002326(n)+A007814(2*(1+n)))^2) - A002326(n) - 3*A007814(2*(1+n)));
write_to_bfile(0,rgs_transform(vector(32769,n,A292268(n-1))),"b292267_upto32768.txt");

A296243 Numbers k such that the multiplicative order of 2 modulo k is even.

Original entry on oeis.org

3, 5, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 33, 35, 37, 39, 41, 43, 45, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 75, 77, 81, 83, 85, 87, 91, 93, 95, 97, 99, 101, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 153

Offset: 1

Max Alekseyev, Dec 09 2017

nonn

Odd numbers k such that A007733(k) = A002326((k-1)/2) is even.

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

Set difference of A005408 and A036259.
Contains A296244 as a subsequence.
The prime terms are given by A014662.
Cf. A002326, A007733.

A036259 = Select[Range[1, 199, 2], OddQ[MultiplicativeOrder[2, #]] &];
Range[1, A036259[[-1]], 2] ~Complement~ A036259 (* Jean-François Alcover, Dec 20 2017 *)
Select[Range[1, 153, 2], EvenQ[MultiplicativeOrder[2, #]] &] (* Amiram Eldar, Jul 30 2020 *)

{ is_A296243(n) = (n%2) && !(znorder(Mod(2,n))%2); }

A302141 Multiplicative order of 16 mod 2n+1.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 3, 1, 2, 9, 3, 11, 5, 9, 7, 5, 5, 3, 9, 3, 5, 7, 3, 23, 21, 2, 13, 5, 9, 29, 15, 3, 3, 33, 11, 35, 9, 5, 15, 39, 27, 41, 2, 7, 11, 3, 5, 9, 12, 15, 25, 51, 3, 53, 9, 9, 7, 11, 3, 6, 55, 5, 25, 7, 7, 65, 9, 9, 17, 69, 23, 15, 7, 21, 37, 15, 6, 5, 13, 13, 33, 81, 5, 83, 39, 9, 43, 15, 29, 89, 45, 15, 9, 10, 9, 95, 24, 3, 49, 99, 33

Offset: 0

Jianing Song, Apr 02 2018

nonn

Reptend length of 1/(2n+1) in hexadecimal.
a(n) <= n; it appears that equality holds if and only if n=1 or is in A163778. - Robert Israel, Apr 02 2018
From Jianing Song, Dec 24 2022: (Start)
a(n) <= psi(2*n+1)/2 <= n. a(n) = psi(2*n+1)/2 if and only if the multiplicative order of 2 modulo 2*n+1 is psi(2*n+1) or psi(2*n+1)/2, and psi(2*n+1) == 2 (mod 4).
a(n) = n if and only if A053447(n) = n and A053447(n) is odd. As a result, a(n) = n if and only if 2*n+1 = p is a prime congruent to 3 modulo 4, and the multiplicative order of 2 modulo p is p-1 or (p-1)/2 (p-1 if p == 3 (mod 8), (p-1)/2 if p == 7 (mod 8)). Such primes p are listed in A105876. (End)

			The fraction 1/13 is equal to 0.13B13B... in hexadecimal, so a(6) = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Jianing Song)
Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326, A053447, A053451, A105876, A163778.

List([0..100],n->OrderMod(16,2*n+1)); # Muniru A Asiru, Feb 25 2019

[1] cat [ Modorder(16, 2*n+1): n in [1..100]]; // Vincenzo Librandi, Apr 03 2018

seq(numtheory:-order(16, 2*n+1), n=0..100); # Robert Israel, Apr 02 2018

Table[MultiplicativeOrder[16, 2 n + 1], {n, 0, 150}] (* Vincenzo Librandi, Apr 03 2018 *)

a(n) = znorder(Mod(16, 2*n+1)) \\ Felix Fröhlich, Apr 02 2018

a(n) = A002326(n)/gcd(A002326(n),4) = A053447(n)/gcd(A053447(n),2). [Corrected by Jianing Song, Dec 24 2022]

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

Original entry on oeis.org

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

A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.

Original entry on oeis.org

4, 12, 16, 48, 80, 112, 132, 208, 240, 1104, 1456, 1892, 2128, 4144, 5852, 12208, 17292, 18544, 21424, 25456, 30160, 45904, 78736, 97552, 106384, 138864, 153596, 154960, 160528, 289772, 311920, 321904, 399212, 430652, 545584, 750064, 770704, 979916, 1037040, 1058512

Offset: 1

Thomas Ordowski, Dec 05 2019

nonn

The condition e(k) > 1 excludes primes and Carmichael numbers.
Numbers n such that e(k) > 1 and b^k == b^e(k) (mod k) for all b.
These are numbers k such that A276976(k) = e(k) > 1.
Are there infinitely many such numbers? Are all such numbers even?
A number k is a term if and only if k is e(k)-Knödel number with e(k) > 1. So they may have the name nonsquarefree e(k)-Knodel numbers k.
It seems that if k is in this sequence, then e(k) = A007814(k) and k/2^e(k) is squarefree.
Conjecture: there are no composite numbers m > 4 such that m == e(m) (mod phi(m)). By Lehmer's totient conjecture, there are no such squarefree numbers.
Problem: are there odd numbers n such that e(n) > 1 and n == e(n) (mod ord_{n}(2)), where ord_{n}(2) = A002326((n-1)/2)? These are odd numbers n such that 2^n == 2^e(n) (mod n) with e(n) > 1.
Numbers k for which A051903(k) > 1 and A219175(k) = A329885(k). - Antti Karttunen, Dec 11 2019

			The number 4 = 2^2 is a term, because e(4) = A051903(4) = 2 > 1 and 4 == 2 (mod lambda(4)), where lambda(4) = A002322(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000010, A002322, A002326, A002997, A050990, A050992, A051903, A068494, A219175, A270096, A276976, A324050, A327979, A329885.

A proper subset of A013929.

Select[Range[10^5], (e = Max @@ Last /@ FactorInteger[#]) > 1 && Divisible[# -e, CarmichaelLambda[#]] &] (* Amiram Eldar, Dec 05 2019 *)

isok(n) = ! issquarefree(n) && (Mod(n, lcm(znstar(n)[2])) == vecmax(factor(n)[, 2])); \\ Michel Marcus, Dec 05 2019

More terms from Amiram Eldar, Dec 05 2019

A336503 2-practical numbers: numbers m such that the polynomial x^m - 1 has a divisor of every degree <= m in the prime field F_2[x].

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 45, 48, 54, 56, 60, 63, 64, 70, 72, 80, 84, 90, 96, 100, 105, 108, 112, 120, 124, 126, 128, 132, 135, 136, 140, 144, 147, 150, 154, 156, 160, 162, 165, 168, 176, 180, 182, 186, 189, 192

Offset: 1

Amiram Eldar, Jul 23 2020

nonn

For a rational prime number p, a "p-practical number" is a number m such that the polynomial x^m - 1 has a divisor of every degree <= m in F_p[x], the prime field of order p.
A number m is 2-practical if and only if every number 1 <= k <= m can be written as Sum_{d|m} A007733(d) * n_d, where A007733(d) is the multiplicative order of 2 modulo the odd part of d, and 0 <= n_d <= phi(d)/A007733(d).
The number of terms not exceeding 10^k for k = 1, 2, ... are 6, 34, 243, 1790, 14703, 120276, 1030279, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Lola Thompson, On the degrees of divisors of T^n-1>, New York Journal of Mathematics, Vo. 19 (2013), pp. 91-116, preprint, arXiv:1206.2084 [math.NT], 2012.
Lola Thompson, Products of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.
Lola Thompson, On the divisors of x^n - 1 in F_p[x], International Journal of Number Theory, Vol. 9, No. 2 (2013), pp. 421-430.
Lola Thompson, Variations on a question concerning the degrees of divisors of x^n - 1, Journal de Théorie des Nombres de Bordeaux, Vol. 26, No. 1 (2014), pp. 253-267.
Eric Weisstein's World of Mathematics, Finite Field.
Wikipedia, Finite field.

Cf. A000265, A002326, A007733, A007814.

Cf. A000010, A260653, A336504, A336505.

rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; mo[n_, p_] := MultiplicativeOrder[p, n/p^IntegerExponent[n, p]]; ppQ[n_, p_] := Module[{d = Divisors[n]}, m = mo[#, p] & /@ d; ns = EulerPhi[d]/m; r = rep[m, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] >  0]; Select[Range[200], ppQ[#, 2] &]

A036260 Numbers k > 1 such that k mod ord2(k) is even, where ord2(k) is the order of 2 mod k.

Original entry on oeis.org

2921, 3017, 3473, 3479, 5767, 5969, 6167, 6377, 6497, 6913, 7223, 7519, 7567, 7751, 9017, 9271, 10199, 10447, 11431, 11929, 12719, 13439, 13609, 14513, 16583, 17009, 17143, 18631, 18809, 19313, 20737, 21119, 22337, 22351, 22537

Offset: 1

David W. Wilson

nonn

These are all composite since for prime p, ord2(p) | phi(p) = p-1, whence p mod ord2(p) = 1.

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

Cf. A002326, A007733, A036259.

Select[Range[3, 23000, 2], EvenQ[Mod[#, MultiplicativeOrder[2, #]]] &] (* Amiram Eldar, Jul 30 2020 *)

Offset corrected by Amiram Eldar, Jul 30 2020

cp's OEIS Frontend

A137605 Consider the sequence: b(0) = n, and for k >= 1, b(k) = b(k-1)/2 if b(k-1) is even, otherwise b(k) = n - (b(k-1)+1)/2. Then a(n) = m, where m is the smallest index such that b(m) = 1.

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A217469 Multiplicative order of 5 (mod 5*n + 1).

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GAP

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A246717 Numbers of the form 2n - 1 such that A246702(n) = 2.

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A292267 Restricted growth sequence transform of A292268; filter combining multiplicative order of 2 mod 2n+1 and the number of trailing 1's in binary expansion of 2n+1.

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A296243 Numbers k such that the multiplicative order of 2 modulo k is even.

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A302141 Multiplicative order of 16 mod 2n+1.

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GAP

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

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A327295 Numbers k such that e(k) > 1 and k == e(k) (mod lambda(k)), where e(k) = A051903(k) is the maximal exponent in prime factorization of k.