A053006 -id:A053006 - cp's OEIS frontend

A036259 Numbers k such that the multiplicative order of 2 modulo k is odd.

1, 7, 23, 31, 47, 49, 71, 73, 79, 89, 103, 127, 151, 161, 167, 191, 199, 217, 223, 233, 239, 263, 271, 311, 329, 337, 343, 359, 367, 383, 431, 439, 463, 479, 487, 497, 503, 511, 529, 553, 599, 601, 607, 623, 631, 647, 713, 719, 721, 727, 743, 751

Offset: 1

David W. Wilson

nonn

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

Closed under multiplication. - Emmanuel Vantieghem, May 07 2025

			2^3 = 1 mod 7, 3 is odd, so 7 is in the sequence.

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

Cf. A002326, A007733, A036260, A053006.

Select[Range[1, 999, 2], OddQ[MultiplicativeOrder[2, #]]&] (* Jean-François Alcover, Dec 20 2017 *)

is(n)=n%2 && znorder(Mod(2,n))%2 \\ Charles R Greathouse IV, Jun 24 2015

from sympy import n_order
from itertools import count, islice
def A036259_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n_order(2,n)&1,count(max(startvalue,1)|1,2))
A036259_list = list(islice(A036259_gen(),20)) # Chai Wah Wu, Feb 07 2023

A291755 Compound filter (multiplicative order of 2 mod 2n+1 & eulerphi(2n+1)): a(n) = P(A002326(n), A037225(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 25, 31, 61, 181, 265, 59, 261, 613, 142, 507, 761, 613, 1513, 566, 416, 607, 2521, 607, 1731, 1499, 607, 2301, 1912, 749, 5305, 1731, 1396, 6613, 7081, 826, 1723, 8581, 2102, 5391, 3169, 1731, 3946, 6709, 5725, 13285, 2493, 3431, 4764, 3415, 2356, 5707, 10201, 3946, 19801, 11527

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000010, A000027, A002326, A037225, A291766 (rgs-version of this filter).
Cf. also A292249, A292268.
Cf. A037226, A053006.

A002326[n_] := MultiplicativeOrder[2, 2n+1];
a[n_] := (1/2)*(2 + ((A002326[n] + EulerPhi[2n+1])^2) - A002326[n] - 3* EulerPhi[2n+1]);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 23 2024 *)

A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
A291755(n) = (1/2)*(2 + ((A002326(n)+eulerphi(n+n+1))^2) - A002326(n) - 3*eulerphi(n+n+1));

(define (A291755 n) (* 1/2 (+ (expt (+ (A002326 n) (A000010 (+ 1 n n))) 2) (- (A002326 n)) (- (* 3 (A000010 (+ 1 n n)))) 2)))

a(n) = (1/2)*(2 + ((A002326(n) + A000010(2n+1))^2) - A002326(n) - 3*A000010(2n+1)).

A291766 Restricted growth sequence transform of A291755; filter combining multiplicative order of 2 mod 2n+1 & eulerphi(2n+1) (A002326 & A037225).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 10, 14, 15, 16, 17, 18, 17, 19, 20, 17, 21, 22, 23, 24, 19, 25, 26, 27, 28, 29, 30, 31, 32, 33, 19, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 34, 45, 46, 29, 47, 48, 43, 49, 50, 41, 51, 52, 53, 45, 54, 55, 56, 57, 43, 58, 59, 60, 61, 49, 62, 63, 64, 51, 65, 66, 67, 68, 69, 53, 70, 71, 57, 72, 61, 73, 74, 75, 61

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

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

Cf. A000010, A002326, A037225, A291755.
Cf. A291769, A292267 for related filters.
Cf. A037226, A053006.

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
A291755(n) = (1/2)*(2 + ((A002326(n)+eulerphi(n+n+1))^2) - A002326(n) - 3*eulerphi(n+n+1));
write_to_bfile(0,rgs_transform(vector(32769,n,A291755(n-1))),"b291766_upto32768.txt");

A117548 Values of n for which there exist d(1),...,d(n), each in {0,1,2} and an r in {1,2} such that Sum_{i=1..n-k} d(i)*d(i+k) == r (mod 3) for all k=0..n-1. (Such a sequence is called a very(3,r) sequence. See the link.).

Original entry on oeis.org

1, 2, 5, 6, 7, 12, 14, 17, 20, 24

Offset: 1

John W. Layman, Mar 28 2006

Theorem. Let a be a very(3,r) sequence of length n, for r=1 or 2 and let z be a sequence of n-1 0's. Then az(2a) is a very(3,3-r) sequence of length 3n-1, where 2a denotes the sequence {2a(i) mod 3, i=1..n}.

			For the sequence d=112102 we get Sum_{i=1..n-k} d(i)*d(i+k) = {11,5,5,5,2,2} = {2,2,2,2,2,2} (mod 3) for k=0..5, so 6 is a term of the sequence.

John W. Layman, On A Generalization of Very Odd Sequences

Cf. A053006, A117549.

A117549 Values of n for which there exist d(1),...,d(n), each in {0,1,...,4} and an r in {1,...,4} such that Sum[d(i)d(i+k),i=1,n-k]=r (mod 5) for all k=0,...,n-1. (Such a sequence is called a very(5,r) sequence. See the link.).

Original entry on oeis.org

1, 3, 6, 10, 13, 16

Offset: 1

John W. Layman, Apr 21 2006

Conjecture. Let A be a very(5,1) (respectively very(5,4)) sequence of length n and let Z be a sequence of n-1 0's.. Then AZ(3A)ZA is a very(5,1) (respectively very(5,4)) sequence of length 5n-2. (Here 3A denotes the result of multiplying each term of A by 3, then reducing modulo 5; and juxtaposition of symbols denotes concatenation of sequences.)

John W. Layman, On A Generalization of Very Odd Sequences

Cf. A053006, A117548.

A360388 Positive integers with binary expansion (b(1), ..., b(m)) such that Sum_{i = 1..m-k} b(i)*b(i+k) is odd for all k = 0..m-1.

Original entry on oeis.org

1, 11, 13, 2787, 3189, 36783, 37063, 43331, 47803, 49813, 56669, 58121, 62961, 9205487, 16215601, 23070091, 23248907, 27264653, 27475981, 43469906355, 55167946629, 75985591407, 80056245671, 81489328999, 83389490039, 87235136243, 88437433811, 90400346819

Offset: 1

Rémy Sigrist, Feb 05 2023

Leading zeros in binary expansions are ignored.
All terms are odd and odious (A092246).
This sequence is infinite since we can, from a given term, build another larger term (see Guy reference).
See A053006 for the distinct binary lengths.
If m is a term, then A030101(m) is also a term.

			For n = 11:
- the binary expansion of 11 is b = (1,1,0,1),
- b(1)*b(1) + b(2)*b(2) + b(3)*b(3) + b(4)*b(4) = 1 + 1 + 0 + 1 = 3 is odd,
- b(1)*b(2) + b(2)*b(3) + b(3)*b(4) = 1 + 0 + 0 = 1 is odd,
- b(1)*b(3) + b(2)*b(4) = 0 + 1 = 1 is odd,
- b(1)*b(4) = 1 is odd,
- so 11 belongs to the sequence.

R. K. Guy, Unsolved Problems in Number Theory, E38.

Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

Cf. A030101, A053006, A092246.

PARI
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See Links section.
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from itertools import count, islice
from functools import reduce
from operator import ixor
def A360388_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        b = tuple(int(d) for d in bin(n)[2:])
        m = len(b)
        if all(reduce(ixor, (b[i]&b[i+k] for i in range(m-k))) for k in range(m)):
            yield n
A360388_list = list(islice(A360388_gen(),10)) # Chai Wah Wu, Feb 07 2023

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A036259 Numbers k such that the multiplicative order of 2 modulo k is odd.

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A291755 Compound filter (multiplicative order of 2 mod 2n+1 & eulerphi(2n+1)): a(n) = P(A002326(n), A037225(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A291766 Restricted growth sequence transform of A291755; filter combining multiplicative order of 2 mod 2n+1 & eulerphi(2n+1) (A002326 & A037225).

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A117548 Values of n for which there exist d(1),...,d(n), each in {0,1,2} and an r in {1,2} such that Sum_{i=1..n-k} d(i)*d(i+k) == r (mod 3) for all k=0..n-1. (Such a sequence is called a very(3,r) sequence. See the link.).

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A117549 Values of n for which there exist d(1),...,d(n), each in {0,1,...,4} and an r in {1,...,4} such that Sum[d(i)d(i+k),i=1,n-k]=r (mod 5) for all k=0,...,n-1. (Such a sequence is called a very(5,r) sequence. See the link.).

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A360388 Positive integers with binary expansion (b(1), ..., b(m)) such that Sum_{i = 1..m-k} b(i)*b(i+k) is odd for all k = 0..m-1.

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