A165781 -id:A165781 - cp's OEIS frontend

A165783 a(n) = A002326(n-1) + A000120(A165781(n-1)).

2, 3, 6, 4, 9, 15, 18, 5, 12, 27, 8, 15, 30, 27, 42, 6, 15, 17, 54, 16, 30, 21, 17, 32, 31, 10, 78, 28, 27, 87, 90, 7, 18, 99, 33, 49, 12, 29, 45, 56, 81, 123, 10, 39, 15, 16, 13, 50, 72, 45, 150, 74, 16, 159, 54, 50, 42, 63, 15, 33, 165, 26, 150, 8, 21, 195, 26, 53, 102, 207

Offset: 1

Ctibor O. Zizka, Sep 26 2009

Given a shift register : r(k)=r(k-1)+ X if r(k-1) is not divisible Y, else r(k)=r(k-1)/Y.
Gcd(r(0), X))=1, Gcd(X, Y)=1.
Then the length of the period orbit of such a register is L + digitsum (r(L)*(Y^L-1)/ X). Digitsum(z)in base X.
r(L) a point from period orbit, L minimal possible exponent such that (Y^L-1)/X)is a positive integer.
Number of period orbits is the order of the cyclic group connected to the register.
a(n) is the period length for Y=2, X=2*n-1, r(L)=1. [Ctibor O. Zizka, Nov 24 2009]

			n=1, a(1)=1 + digitsum(1)= 2.
n=2, a(2)=2 + digitsum(1)=3.
n=3, a(3)= 4 + digitsum(3) = 6.
n=4, a(4)= 3 + digitsum(1)=4.
n=5, a(5)= 6 + digitsum(7)=9. [_Ctibor O. Zizka_, Nov 24 2009]

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

Cf. A002326, A053446.

A002326 := proc(n) if n = 0 then 1; else numtheory[order](2,2*n+1) ; end if ; end proc:
A165781 := proc(n) (2^A002326(n)-1)/(2*n+1) ; end proc:
read("transforms") ; A165783 := proc(n) A002326(n-1)+wt(A165781(n-1) ) ; end proc:
seq(A165783(n),n=1..80) ; # R. J. Mathar, Nov 26 2009

Table[(b = MultiplicativeOrder[2, 2 n - 1]) + Plus @@ IntegerDigits[(2^b - 1)/(2 n - 1), 2], {n, 1, 70}] (* Ivan Neretin, May 09 2015 *)

hamming(n)=my(v=binary(n));sum(i=1,#v,v[i])
a(n)=my(x=2*n+1,m=znorder(Mod(2,x)));m+hamming((1<

a(n) = L + digitsum((2^L -1)/(2*n-1)). Digitsum(z)in base 2. [Ctibor O. Zizka, Nov 24 2009]

Program and extension by Charles R Greathouse IV, Nov 24 2009

Definition corrected and comments merged by R. J. Mathar, Nov 26 2009

A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime.

Original entry on oeis.org

21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, 417, 429, 453, 465, 483, 489, 495, 497, 505, 507, 525, 543, 555, 579, 597, 605, 609, 615, 627, 633, 651, 655

Offset: 1

Felix Fröhlich, Apr 23 2012

nonn

The primes in this sequence are A001220, the Wieferich primes. - Charles R Greathouse IV, Feb 02 2014

Odd prime p is a Wieferich prime if and only if A002326((p^2-1)/2) = A002326((p-1)/2). See the sixth comment to A001220 and my formula below. - Thomas Ordowski, Feb 03 2014

			21 is in the sequence because the multiplicative order of 2 mod 21 is 6, and (2^6-1)/21 = 3, which is not coprime to 21.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Z. Franco and C. Pomerance, On a conjecture of Crandall concerning the qx + 1 problem, Math. Comp. Vol. 64, No. 211 (1995), 1333-1336.

For another definition of Wieferich numbers, see A077816.

Cf. A002326.

with(numtheory):
a:= proc(n) option remember; local q;
      for q from 2 +`if`(n=1, 1, a(n-1)) by 2
        while igcd((2^order(2, q)-1)/q, q)=1 do od; q
    end:
seq (a(n), n=1..60);  # Alois P. Heinz, Apr 23 2012

Select[Range[1, 799, 2], GCD[#, (2^MultiplicativeOrder[2, #] - 1)/#] > 1 &] (* Alonso del Arte, Apr 23 2012 *)

is(n)=n%2 && gcd(lift(Mod(2,n^2)^znorder(Mod(2,n))-1)/n,n)>1 \\ Charles R Greathouse IV, Feb 02 2014

Odd numbers q such that A002326((q^2-1)/2) < q * A002326((q-1)/2). Other positive odd integers satisfy the equality. - Thomas Ordowski, Feb 03 2014

Odd numbers q such that gcd(A165781((q-1)/2), q) > 1. - Thomas Ordowski, Feb 12 2014

A249596 Analog of A097717 in base 2.

Original entry on oeis.org

1, 2, 9, 4, 35, 558, 2205, 8, 135, 137970, 33, 1068, 545259, 135926, 138845925, 16, 527, 2106, 35288379945, 2100, 537075, 8382, 2093, 4283544, 1069975, 130, 2294286602622705, 533820, 133371, 146557818382226310, 585910928570692725, 32, 2079

Offset: 1

R. J. Mathar, Mar 30 2009

Conjecture: a(n) = n*A165781(n). - R. J. Mathar, Nov 11 2014

Lars Blomberg, Table of n, a(n) for n = 1..1661

Cf. A094676, A249597, A249598, A249599.

A249596 := proc(n)
    local m,b,mbas,msf ;
    b := 2;
    for m from 1 to 1999999 do
        mbas := convert(m,base,b) ;
        msf := [op(-1,mbas),op(1..nops(mbas)-1,mbas)] ;
        msf := add(op(i,msf)*b^(i-1),i=1..nops(msf)) ;
        if m/n = msf then
            return m;
        end if;
    end do:
    -1 ;
end proc:
for n from 1 do
    print(n,A249596(n)) ;
end do: # R. J. Mathar, Nov 11 2014

a(15)-a(33) from Lars Blomberg, Feb 05 2015

A352217 Smallest power of 2 that is one more than a multiple of 2n-1.

Original entry on oeis.org

2, 4, 16, 8, 64, 1024, 4096, 16, 256, 262144, 64, 2048, 1048576, 262144, 268435456, 32, 1024, 4096, 68719476736, 4096, 1048576, 16384, 4096, 8388608, 2097152, 256, 4503599627370496, 1048576, 262144, 288230376151711744, 1152921504606846976, 64, 4096

Offset: 1

J. Lowell, Mar 07 2022

Every odd number is a divisor of a number of the form 2^n-1.

			a(5)=64 because 63 is the smallest number of the form 2^n-1 that's a multiple of 9.

Michael S. Branicky, Table of n, a(n) for n = 1..1661

Cf. A002326, A005408, A165781.

a:= n-> 2^`if`(n=1, 1, numtheory[order](2, 2*n-1)):
seq(a(n), n=1..50);  # Alois P. Heinz, Mar 07 2022

Table[2^MultiplicativeOrder[2, 2*n - 1], {n, 1, 33}] (* Amiram Eldar, Mar 08 2022 *)

a(n) = 1 << znorder(Mod(2,2*n-1)); \\ Kevin Ryde, Mar 07 2022

def a(n):
    if n == 1: return 2
    p, m = 2, 2*n-1
    while p <= m or p % m != 1: p *= 2
    return p
print([a(n) for n in range(1, 34)]) # Michael S. Branicky, Mar 07 2022

from sympy import n_order
def a(n): return 2**n_order(2, 2*n-1)
print([a(n) for n in range(1, 34)]) # Michael S. Branicky, Mar 07 2022 after Alois P. Heinz

From Alois P. Heinz, Mar 07 2022: (Start)
a(n) = 2^A002326(n-1).
a(n) = 1 + A165781(n-1)*(2*n-1). (End)

a(14) and beyond from Michael S. Branicky, Mar 07 2022

A297362 Numbers k such that (2^ord(2, k) - 1)/k is prime, where ord(2, k) is the multiplicative order of 2 (mod k).

Original entry on oeis.org

5, 9, 21, 23, 33, 47, 51, 73, 85, 89, 93, 129, 167, 217, 223, 263, 315, 341, 381, 585, 819, 1057, 1365, 3591, 3855, 4681, 4871, 5461, 6141, 6223, 6719, 7487, 8193, 11447, 13107, 13367, 13797, 14329, 16513, 18631, 21845, 24573, 25575, 26431, 33825, 37449

Offset: 1

Amiram Eldar, Dec 29 2017

nonn

The corresponding primes are 3, 7, 3, 89, 31, 178481, 5, 7, 3, 23, 11, ...

			5 is in the sequence since ord(2, 5) = 4 and (2^4 - 1)/5 = 3 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..64

Cf. A002326, A165781.

aQ[n_] := PrimeQ[(2^MultiplicativeOrder[2, n] - 1)/n]; Select[Range[10000],aQ]

is(n) = n%2 && isprime((2^znorder(Mod(2, n))-1)/n); \\ Amiram Eldar, Aug 26 2023

A237663 Odd numbers m such that the order of 2 mod m^3 is less than m times the order of 2 mod m^2.

Original entry on oeis.org

57, 111, 219, 285, 327, 399, 489, 505, 543, 555, 597, 627, 741, 777, 813, 969, 1083, 1095, 1137, 1221, 1255, 1299, 1311, 1379, 1425, 1443, 1461, 1467, 1515, 1533, 1569, 1623, 1635, 1653, 1731, 1767, 1839, 1887, 1893, 1995, 2005, 2109, 2271, 2289, 2337, 2409, 2433, 2445, 2451, 2487, 2553, 2649, 2679, 2715, 2757, 2775, 2793, 2811, 2847, 2973, 2985, 3005, 3021, 3027, 3135, 3189, 3219, 3351, 3363, 3423, 3437, 3441, 3459, 3477, 3505, 3513

Offset: 1

Thomas Ordowski and Charles R Greathouse IV, Feb 11 2014

nonn

These numbers m are a subset of the {A182297} Wieferich numbers (2).

All known numbers m are composite. A prime p satisfies this inequality if and only if the order of 2 mod p^3 is the order of 2 mod p, which is equivalent to p^3 dividing 2^(p-1)-1, but no such prime p are known (as opposed to the A001220 Wieferich primes).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002326, A182297.

okQ[m_] := MultiplicativeOrder[2, m^3] < m*MultiplicativeOrder[2, m^2]; Select[Range[1, 9999, 2], okQ] (* Jean-François Alcover, Dec 10 2015 *)

is(m)=m%2 && znorder(Mod(2, m^3)) < m*znorder(Mod(2, m^2))

Odd numbers m such that A002326((m^3-1)/2) < m * A002326((m^2-1)/2).

Odd numbers m such that 1 < gcd(A165781((m-1)/2), m) is a square.

cp's OEIS Frontend

A165783 a(n) = A002326(n-1) + A000120(A165781(n-1)).

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A182297 Wieferich numbers (2): positive odd integers q such that q and (2^A002326((q-1)/2)-1)/q are not relatively prime.

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A249596 Analog of A097717 in base 2.

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A352217 Smallest power of 2 that is one more than a multiple of 2n-1.

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A297362 Numbers k such that (2^ord(2, k) - 1)/k is prime, where ord(2, k) is the multiplicative order of 2 (mod k).

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A237663 Odd numbers m such that the order of 2 mod m^3 is less than m times the order of 2 mod m^2.

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