A002326 -id:A002326 - cp's OEIS frontend

A292249 Compound filter (multiplicative order of 2 mod 2n+1 & prime signature of 2n+1): a(n) = P(A002326(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

1, 5, 14, 9, 42, 65, 90, 40, 44, 189, 61, 77, 273, 318, 434, 20, 115, 148, 702, 148, 230, 119, 265, 299, 297, 86, 1430, 320, 271, 1769, 1890, 142, 148, 2277, 373, 665, 54, 485, 625, 819, 2400, 3485, 86, 556, 77, 148, 115, 856, 1224, 850, 5150, 1377, 832, 5777, 702, 856, 434, 1220, 265, 430, 6438, 320, 5771, 35, 185, 8645, 271

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

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

Cf. A000027, A002326, A046523, A278223, A286573, A291769 (rgs-version of the same filter).

Cf. also A291755, A292268.

A002326(n) = if(n<0, 0, znorder(Mod(2, 2*n+1))); \\ This function from Michael Somos, Mar 31 2005
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A292249(n) = (1/2)*(2 + ((A002326(n)+A046523(n+n+1))^2) - A002326(n) - 3*A046523(n+n+1));

(define (A292249 n) (* 1/2 (+ (expt (+ (A002326 n) (A046523 (+ 1 n n))) 2) (- (A002326 n)) (- (* 3 (A046523 (+ 1 n n)))) 2)))

a(n) = (1/2)*(2 + ((A002326(n) + A046523(2n+1))^2) - A002326(n) - 3*A046523(2n+1)).

A292266 Restricted growth sequence transform of A292265; a filter related to Shevelev's algorithm for computing A002326.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 12, 10, 13, 14, 15, 16, 17, 16, 18, 19, 16, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 24, 34, 35, 36, 37, 38, 39, 11, 16, 6, 40, 41, 42, 43, 44, 7, 45, 46, 47, 48, 49, 50, 51, 52, 53, 43, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 57, 76, 77, 78, 79, 80, 81

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

For all i, j: a(i) = a(j) => A002326(i) = A002326(j).

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

Cf. A002326, A292239, A292265.

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])); }
A000265(n) = (n >> valuation(n, 2));
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A292265(n) = { my(x = n+n+1, z = A019565(valuation(1+x,2)), m = A000265(1+x)); while(m!=1, z *= A019565(valuation(x+m,2)); m = A000265(x+m)); z; };
write_to_bfile(0,rgs_transform(vector(32769,n,A292265(n-1))),"b292266_upto32768.txt");

A292268 Compound filter (multiplicative order of 2 mod 2n+1 & number of trailing 1's in binary expansion of 2n+1): a(n) = P(A002326(n), A007814(2n+2)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 10, 13, 21, 65, 78, 25, 36, 189, 21, 89, 210, 189, 406, 41, 55, 90, 666, 103, 210, 119, 78, 348, 231, 44, 1378, 251, 171, 1769, 1830, 61, 78, 2277, 253, 701, 45, 230, 465, 900, 1485, 3485, 36, 463, 66, 90, 55, 816, 1176, 495, 5050, 1429, 78, 5777, 666, 777, 406, 1034, 78, 349, 6105, 230, 5050, 85, 105, 8645, 171, 739, 2346, 9729, 1081

Offset: 0

Antti Karttunen, Oct 02 2017

nonn

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

Cf. A000027, A002326, A007814, A292267 (rgs-version of this filter).

Cf. also A291755, A292249.

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)));

(define (A292268 n) (* 1/2 (+ (expt (+ (A002326 n) (A007814 (+ 2 n n))) 2) (- (A002326 n)) (- (* 3 (A007814 (+ 2 n n)))) 2)))

a(n) = (1/2)*(2 + ((A002326(n) + A007814(2n+2))^2) - A002326(n) - 3*A007814(2n+2)).

A179460 Numbers m for which 2*A179382(m)=A002326(m-1).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 13, 14, 15, 17, 19, 21, 22, 27, 29, 30, 31, 33, 34, 35, 39, 41, 42, 49, 50, 51, 54, 55, 57, 61, 63, 65, 66, 69, 70, 71, 73, 75, 79, 82, 85, 86, 87, 89, 90, 91, 93, 97, 99, 101, 102, 103, 104, 105, 106, 107, 114, 115, 121, 122, 125, 126, 129, 133, 135

Offset: 1

Vladimir Shevelev, Jul 14 2010

nonn

m is in the sequence iff the set {1,2,...,2^(2*m-2)} considered in reduced residue system modulo 2*m-1 contains the same number of odd and even integers.

			5 in the sequence since modulo 2*5-1=9 we have {1,2,4,8,16,32}={1,2,4,8,7,5} and the last set contains 3 odd and 3 even elements.

Cf. A002326, A179382.

fQ[n_] := Block[{r = Union@ PowerMod[2, Range[0, 2 n - 2], 2 n - 1]}, Length@ r == 2 Count[ OddQ@ r, True]]; Select[ Range@ 138, fQ] (* Robert G. Wilson v, Aug 26 2010 *)

More terms from Robert G. Wilson v, Aug 26 2010

A274298 A bisection of A002326.

Original entry on oeis.org

1, 4, 6, 12, 8, 6, 20, 28, 10, 36, 20, 12, 21, 52, 18, 60, 12, 22, 9, 30, 54, 8, 11, 10, 48, 100, 12, 36, 28, 12, 110, 100, 14, 18, 68, 46, 28, 148, 24, 52, 33, 20, 156, 172, 58, 180, 36, 18, 96, 196, 66, 20, 90, 70, 15, 24, 60, 76, 29, 78, 24, 84, 82, 110, 16, 84, 52, 268, 12, 92, 70, 36, 136, 292, 90

Offset: 0

N. J. A. Sloane, Jun 20 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A002326.

A274299 A bisection of A002326.

Original entry on oeis.org

2, 3, 10, 4, 18, 11, 18, 5, 12, 12, 14, 23, 8, 20, 58, 6, 66, 35, 20, 39, 82, 28, 12, 36, 30, 51, 106, 36, 44, 24, 20, 7, 130, 36, 138, 60, 42, 15, 20, 52, 162, 83, 18, 60, 178, 60, 40, 95, 12, 99, 84, 66, 210, 28, 18, 37, 226, 30, 92, 119, 162, 36, 50, 8, 36, 131, 22, 135, 20, 30, 94, 60, 48, 116

Offset: 0

N. J. A. Sloane, Jun 20 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, arXiv:1412.8533 [math.CO], 2014.
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, The American Mathematical Monthly 123.6 (2016): 542-556.

Cf. A002326, A007346, A274298.

Table[MultiplicativeOrder[2, 4 n + 3], {n, 0, 73}] (* Stan Wagon, Jun 24 2016 *)

A139791 Numbers n for which 2n is a multiple of A002326(n), the multiplicative order of 2 mod 2n+1.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 170

Offset: 1

Vladimir Shevelev, May 21 2008, May 24 2008

nonn

The sequence properly contains A005097. 170 is the first number which is not in A005097. One can prove that A002326(2^(2t-1)) = 4t. Thus if n=2^(2t-1), where, for any m>0, t=2^(m-1) then 2n is a multiple of A002326(n) while 2n+1 is a Fermat number which, as well known, is not always a prime.

The sequence is the union of A005097 and (A001567 - 1)/2. [Conjectured by Vladimir Shevelev, proved by Ray Chandler, May 26 2008]

Christopher Adler and Jean-Paul Allouche (2022), Finite self-similar sequences, permutation cycles, and music composition, Journal of Mathematics and the Arts, 16:3, 244-261, DOI: 10.1080/17513472.2022.2116745.

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

Cf. A002326, A005097, A001262, A001567, A137576.

Select[Range[160], Divisible[2#, MultiplicativeOrder[2, 2#+1]] &] (* Amiram Eldar, Jun 28 2019 *)

isok(n) = !(2*n % znorder(Mod(2, 2*n+1))); \\ Michel Marcus, Nov 02 2017

Data extended up to a(68) = 170 to clarify distinction from A005097 and essentially identical sequences A130290 and A102781, by M. F. Hasler, Dec 13 2019

A237292 a(n) = A002326(2n(n+1)) / A002326(n).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 7, 23, 25, 27, 29, 31, 33, 35, 37, 13, 41, 43, 45, 47, 49, 51, 53, 11, 19, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 35, 107, 109, 37, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Thomas Ordowski, Feb 06 2014

nonn

Note that ((2n+1)^2-1)/2 = 2n(n+1).
We have 1 <= a(n) <= 2n+1 and a(n) divides 2n+1 for every n >= 0.
Odd m is a Wieferich number A182297 if and only if a((m-1)/2) < m.
Odd prime p is a Wieferich prime A001220 if and only if a((p-1)/2) = 1.
a((n-1)/2) = 1 for n = 1, 1093, 3511, 7651, 10533, 14209, 17555, ...

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001220, A002326, A077816 and A182297.

1,seq(numtheory:-order(2,4*n*(n+1)+1)/numtheory:-order(2,2*n+1),n=1..100); # Robert Israel, Dec 02 2015

a002326(n) = znorder(Mod(2, 2*n+1));
a(n) = a002326(2*n*(n+1))/a002326(n); \\ Michel Marcus, Feb 08 2014

a(n) = ord_{(2n+1)^2}(2) / ord_{2n+1}(2), n >= 0.

More terms from Michel Marcus, Feb 08 2014

Edited by Thomas Ordowski, Dec 02 2015

A359147 Partial sums of A002326.

Original entry on oeis.org

1, 3, 7, 10, 16, 26, 38, 42, 50, 68, 74, 85, 105, 123, 151, 156, 166, 178, 214, 226, 246, 260, 272, 295, 316, 324, 376, 396, 414, 472, 532, 538, 550, 616, 638, 673, 682, 702, 732, 771, 825, 907, 915, 943, 954, 966, 976, 1012, 1060, 1090, 1190, 1241, 1253, 1359, 1395, 1431

Offset: 0

N. J. A. Sloane, Feb 14 2023

nonn

a(n)/n is the average order of 2 mod m, averaged over all odd numbers m from 1 to 2n+1. From Kurlberg-Pomerance (2013), this is of order constant*n/log(n). So the graph of this sequence grows like constant*n^2/log(n). [The asymptotic formula involves the constant B = 0.3453720641..., A218342. - Amiram Eldar, Feb 15 2023]

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Pär Kurlberg and Carl Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra & Number Theory, Vol. 7, No. 4 (2013), pp. 981-999.

Cf. A002326, A007733, A218342.

a:= proc(n) option remember;
     `if`(n=0, 1, a(n-1)+numtheory[order](2, 2*n+1))
    end:
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 14 2023

Accumulate[MultiplicativeOrder[2,#]&/@Range[1,151,2]] (* Harvey P. Dale, Jul 08 2023 *)

a(n) = sum(k = 0, n, if(k<0, 0, znorder(Mod(2, 2*k+1)))) \\ Thomas Scheuerle, Feb 14 2023

from sympy import n_order
def A359147(n): return sum(n_order(2,m) for m in range(1,n+1<<1,2)) # Chai Wah Wu, Feb 14 2023

a(n) = Sum_{k = 0..n} A007733(2*k+1). - Thomas Scheuerle, Feb 15 2023

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

Original entry on oeis.org

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

cp's OEIS Frontend

A292249 Compound filter (multiplicative order of 2 mod 2n+1 & prime signature of 2n+1): a(n) = P(A002326(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

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A292266 Restricted growth sequence transform of A292265; a filter related to Shevelev's algorithm for computing A002326.

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A292268 Compound filter (multiplicative order of 2 mod 2n+1 & number of trailing 1's in binary expansion of 2n+1): a(n) = P(A002326(n), A007814(2n+2)), where P(n,k) is sequence A000027 used as a pairing function.

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A179460 Numbers m for which 2*A179382(m)=A002326(m-1).

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Mathematica

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A274298 A bisection of A002326.

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A274299 A bisection of A002326.

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A139791 Numbers n for which 2n is a multiple of A002326(n), the multiplicative order of 2 mod 2n+1.

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A237292 a(n) = A002326(2n(n+1)) / A002326(n).

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A359147 Partial sums of A002326.

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