A261524 -id:A261524 - cp's OEIS frontend

A261862 Terms in A261524 that are not multiples of earlier terms.

3, 7, 31, 73, 85, 127, 2047, 3133, 4369, 8191, 11275, 49981, 60787, 76627, 121369, 131071, 140911, 178481, 262657, 486737, 524287, 599479, 1082401

Offset: 1

Joerg Arndt, Sep 07 2015

All Mersenne primes >= 3 are terms (see A001348).
From Jianing Song, Oct 13 2023: (Start)
In A261524 it is conjectured that degree(gcd( 1 + x^(Zs(d,2,1)), 1 + (1+x)^(Zs(d,2,1))) > 0 for every odd number d != 1, 15, 21, where Zs(d,2,1) is the d-th Zsigmondy number with parameters (2,1) (A064078). Since Zsigmondy numbers with different indices are coprime, if this conjecture is true, then there exists a term of this sequence k with ord(2,k) = d, and k must be a divisor of Zs(d,2,1) for every odd number d != 1, 15, 21. Here ord(a,k) is the multiplicative order of 2 modulo k. In A261524 we show that this conjecture is true for powers > 1 of a prime r >= 5, so there are infinitely many terms in this sequence.
One may conjecture that, if k is a term with ord(2,k) = d for even d, then k is a divisor of Zs(d,2,1)*Zs(d/2,2,1). This fails for (d,k) = (20,11275), (40,16962275), (44,165965585), ...
Conjecture: a term with ord(2,k) = d for even d exists if and only if d != 12 or 2*p, where p is any Mersonne exponent. (End)

Jianing Song, List of terms k of A261862 with ord(2,k) <= 47

Cf. A261524, A001348.

n=1; t= L= {}; While[n<5000, n+=2; If[ CoefficientList[ PolynomialGCD[1 + x^n, 1 + (x + 1)^n, Modulus->2], x] !={1}, If[ Intersection[Divisors@ n, t] == {}, Print@ AppendTo[L, n]]; AppendTo[t, n]]]; L (* Giovanni Resta, Sep 07 2015 *)

Corrected and extended by Giovanni Resta, Sep 09 2015

Terms a(17)-a(23) from Joerg Arndt, Sep 10 2015

A191131 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x and 4x+3 are in a.

Original entry on oeis.org

1, 3, 7, 9, 15, 21, 27, 31, 39, 45, 63, 81, 87, 93, 111, 117, 127, 135, 159, 183, 189, 243, 255, 261, 279, 327, 333, 351, 375, 381, 405, 447, 471, 477, 511, 543, 549, 567, 639, 729, 735, 759, 765, 783, 837, 975, 981, 999, 1023, 1047, 1053, 1119, 1125, 1143, 1215, 1311, 1335, 1341, 1407, 1413, 1431, 1503, 1527, 1533, 1623

Offset: 1

Clark Kimberling, May 27 2011

nonn

See A191113.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A191113, A191186, A191187.

Note that A191131, A261524, A261871, and A282572 are very similar and easily confused with each other.

import Data.Set (singleton, deleteFindMin, insert)
a191131 n = a191131_list !! (n-1)
a191131_list = f $ singleton 1
   where f s = m : (f $ insert (3*m) $ insert (4*m+3) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 3; i = 0; j = 4; k = 3; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]   (* A191131 *)
b = a/3; c = (a - 3)/4; r = Range[1, 1500];
d = Intersection[b, r] (* A191186 *)
e = Intersection[c, r] (* A191187 *)

A282572 Integers that are a product of Mersenne numbers A000225, (i.e., product of numbers of the form 2^n - 1).

Original entry on oeis.org

1, 3, 7, 9, 15, 21, 27, 31, 45, 49, 63, 81, 93, 105, 127, 135, 147, 189, 217, 225, 243, 255, 279, 315, 343, 381, 405, 441, 465, 511, 567, 651, 675, 729, 735, 765, 837, 889, 945, 961, 1023, 1029, 1143, 1215, 1323, 1395, 1519, 1533, 1575, 1701, 1785, 1905, 1953, 2025, 2047, 2187, 2205, 2295, 2401

Offset: 1

Andrew Ivashenko, Feb 18 2017

nonn

Odd orders of finite abelian groups that appear as the group of units in a commutative ring (Chebolu and Lockridge, see A296241). - Jonathan Sondow, Dec 15 2017

Actually, the Chebolu and Lockridge paper states that this sequence gives all odd numbers that are possible numbers of units in a (commutative or non-commutative) ring (Ditor's theorem). Concretely, if k = (2^(e_1)-1)*(2^(e_2)-1)*...(2^(e_r)-1) is a term, let R = (F_2)^s X F_(2^(e_1)) X F_(2^(e_2)) X ... X F_(2^(e_r)) for s >= 0, then the number of units in R is k. - Jianing Song, Dec 23 2021

			63 = 1*3^3*7, 81 = 1*3^4, 93 = 1*3*31, 105 = 1*7*15, 41013 = 1*3^3*7^2*31.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Sunil K. Chebolu and Keir Lockridge. How many units can a commutative ring have?, arXiv preprint arXiv:1701.02341 [math.AC], 2017. See Th. 8.

Cf. A000225, A056652, A296241.

Note that A191131, A261524, A261871, and A282572 are very similar and easily confused with each other.

d:= 15: # for terms < 2^d
N:= 2^d:
S:= {1}:
for m from 2 to d do
  r:= 2^m-1;
  k:= ilog[r](N);
  V:= S;
  for i from 1 to k do
    V:= select(`<`, map(`*`, V, r), N);
    S:= S union V
  od;
od:
sort(convert(S, list)); # Ridouane Oudra, Sep 14 2021

lmt = 2500; a = b = Array[2^# - 1 &, Floor@ Log2@ lmt]; k = 2; While[k < Length@ a, e = 1; While[e < Floor@ Log[ a[[k]], lmt], b = Union@ Join[b, Select[ a[[k]]^e*b, # < 1 + lmt &]]; e++]; k++]; b (* Robert G. Wilson v, Feb 23 2017 *)

forstep(x=1,1000000,2, t=x; forstep(n=20,2,-1, m=2^n-1; while(t%m==0, t=t\m)); if(t==1, print1(x,","))) \\ Dmitry Petukhov, Feb 23 2017

More terms from Michel Marcus, Feb 23 2017

Definition changed by David A. Corneth, Mar 12 2017

A261871 Numbers of the form (2j-1)(2^k-1); j>=1, k>=2.

Original entry on oeis.org

3, 7, 9, 15, 21, 27, 31, 33, 35, 39, 45, 49, 51, 57, 63, 69, 75, 77, 81, 87, 91, 93, 99, 105, 111, 117, 119, 123, 127, 129, 133, 135, 141, 147, 153, 155, 159, 161, 165, 171, 175, 177, 183, 189, 195, 201, 203, 207, 213, 217, 219, 225, 231, 237, 243, 245, 249, 255, 259, 261, 267, 273, 279, 285, 287, 291, 297, 301

Offset: 1

Bob Selcoe, Sep 04 2015

Odd numbers complementary to A185208.

Lim_{n->inf.} a(n)/n > 6/(1 + Sum_{j>=1} (2/(2^(2j+1)-1))) ~ 4.375745.

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

Cf. A185208.

Note that A191131, A261524, A261871, and A282572 are very similar and easily confused with each other.

lmt = 310; Take[ Union@ Flatten@ Table[ (2j - 1)(2^k - 1), {j, lmt/4}, {k, 2, 1 + Log2[ lmt/(2j)] }], 68] (* Michael De Vlieger, Sep 04 2015 *) (* and modified by Robert G. Wilson v, Sep 05 2015 *)

list(lim)=my(v=List(),t); for(k=2,logint(lim\1+1,2), t=2^k-1; forstep(j=1,lim\t,2, listput(v,t*j))); Set(v) \\ Charles R Greathouse IV, Sep 05 2015

2n < a(n) < 5n. For n > 51, 4.3n < a(n) < 4.5n. - Charles R Greathouse IV, Sep 05 2015

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A261862 Terms in A261524 that are not multiples of earlier terms.

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A191131 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x and 4x+3 are in a.

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A282572 Integers that are a product of Mersenne numbers A000225, (i.e., product of numbers of the form 2^n - 1).

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A261871 Numbers of the form (2*j-1)*(2^k-1); j>=1, k>=2.

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A261871 Numbers of the form (2j-1)(2^k-1); j>=1, k>=2.