A000224 -id:A000224 - cp's OEIS frontend

A118382 Primitive Orloj clock sequences; row n sums to 2n-1.

1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 1, 2, 3, 3, 1, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 2, 3, 1, 2, 3, 4, 3, 2, 1, 1, 1, 1, 2, 4, 1, 4, 2, 1, 1, 1, 3, 1, 2, 1, 5, 2, 2, 1, 2, 3, 1, 3, 3, 2, 6, 1, 2, 2, 1, 3, 1, 3, 2, 5, 1, 1, 1, 1, 2, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 3, 1, 2, 3, 3, 3, 3, 3, 1, 2, 1, 2, 1, 1, 2, 5, 1, 2, 2

Offset: 1

Franklin T. Adams-Watters, Apr 26 2006

An Orloj clock sequence is a finite sequence of positive integers that, when iterated, can be grouped so that the groups sum to successive natural numbers. There is one primitive sequence whose values sum to each odd m; all other sequences can be obtained by repeating and refining these. Refining means splitting one or more terms into values summing to that term. The Orloj clock sequence is the one summing to 15: 1,2,3,4,3,2, with a beautiful up and down pattern.

These are known in some papers as Sindel sequences. It appears that this sequence was submitted prior to the first such publication.

			For a sum of 5, we have 1,2,2, which groups as 1, 2, 2+1, 2+2, 1+2+2, 1+2+2+1, .... This could be refined by splitting the second 2, to give the sequence 1,2,1,1; note that when this is grouped, the two 1's from the refinement always wind up in the same sum.
The array starts:
  1;
  1,2;
  1,2,2;
  1,2,3,1;
  1,2,3,3;
  1,2,1,2,4,1;
  ...

Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.

Cf. A028355, A118383. Length of row n is A117484(2n-1) = A000224(2n-1).

{Orloj(n) = local(found,tri,i,last,r); found = vector(n,i,0); found[n] = 1; tri = 0; for(i = 1, if(n%2==0,n-1,n\2), tri += i; if(tri >= n, tri -= n); found[tri] = 1); last = 0; r = []; for(i = 1, n, if(found[i], r = concat(r, [i-last]); last = i)); r}

Let b(i),0<=i

A084848 a(n) is the number of quadratic residues of A085635(n).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 7, 8, 12, 14, 16, 16, 24, 28, 32, 42, 48, 48, 48, 64, 84, 96, 112, 144, 144, 176, 192, 192, 288, 336, 336, 504, 576, 576, 704, 864, 1008, 1056, 1152, 1232, 1152, 1344, 1728, 1920, 2016, 2016, 2352

Offset: 1

Jose R. Brox (tautocrona(AT)terra.es), Jul 12 2003

nonn

Note that the terms are not all distinct.

			a(2)=2 because there are 2 different quadratic residues modulo 3, so 3 has 66.67% of quadratic residues density, while 2 has a 100%, so 3 has the least quadratic residues density up to 3.

Keith F. Lynch, Table of n, a(n) for n = 1..200 (first 111 terms from Hugo Pfoertner).
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.

Cf. A000224, A085635.

Block[{s = Range[0, 2^15 + 1]^2, t}, t = Array[{#1/#2, #2} & @@ {#, Length@ Union@ Mod[Take[s, # + 1], #]} &, Length@ s - 1]; Map[t[[All, -1]][[FirstPosition[t[[All, 1]], #][[1]] ]] &, Union@ FoldList[Max, t[[All, 1]] ] ] ] (* Michael De Vlieger, Sep 10 2018 *)

a000224(n)=my(f=factor(n));prod(i=1,#f[,1],if(f[i,1]==2,2^f[1,2]\6+2,f[i,1]^(f[i,2]+1)\(2*f[i,1]+2)+1)) \\ from Charles R Greathouse IV
r=2;for(k=1,1e6,v=a000224(k);t=v/k;if(tHugo Pfoertner, Aug 24 2018

a(n) = A000224(A085635(n)). - Hugo Pfoertner, Aug 24 2018

More terms from Jud McCranie, Jul 18 2003

a(1) corrected by Hugo Pfoertner, Aug 23 2018

A117484 Number of triangular numbers mod n.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 4, 8, 4, 6, 6, 8, 7, 8, 6, 16, 9, 8, 10, 12, 8, 12, 12, 16, 11, 14, 11, 16, 15, 12, 16, 32, 12, 18, 12, 16, 19, 20, 14, 24, 21, 16, 22, 24, 12, 24, 24, 32, 22, 22, 18, 28, 27, 22, 18, 32, 20, 30, 30, 24, 31, 32, 16, 64, 21, 24, 34, 36, 24, 24, 36, 32, 37, 38, 22

Offset: 1

Franklin T. Adams-Watters, Apr 25 2006

Same as A000224 (number of squares mod n) for n odd, since there we can divide by 2 and then complete the square.
From David Morales Marciel, Jul 13 2015: (Start)
a(n) is also the total number of vertices of an n-gon that are a "final vertex" of a bouncing pattern representing the modulo-n series (an image of the bouncing pattern is included in the LINKS section). It is defined by the following algorithm:
(1) Define counter c=1.
(2) Start at any desired vertex.
(3) Mark the current vertex as a "final vertex".
(4) Advance clockwise c vertices.
(5) Set c=c+1.
(6) Repeat from (3).
The pattern of "final vertices" is cyclic: after some repetitions of steps (3)-(6), the marking of vertices is repeated and it is possible to count how many vertices of the n-gon contain a "final vertex" mark.
Examples: trivial case: a(n)=1 (one vertex is always a "final vertex"). From that point following the algorithm: a(2)=2 (segment, both vertices are a "final vertex"), a(3)=2 (triangle, only two vertices are "final vertex"), etc.
(End)

			When n=3, there is no triangular number which is congruent to 2 (mod 3) but only == 0 or 1 (mod 3), so a(3) = 2. - _Robert G. Wilson v_, Sep 16 2015

Robert Israel, Table of n, a(n) for n = 1..10000
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 21.
David Morales Marciel, Bouncing patterns of some modulo-n series (n-gons 3 to 14, 16, 17 and 27), according to the algorithm explained in the comments. Current sequence belongs to the bouncing pattern of the 3-gon sample (first at the top).

Cf. A000224.

a:= proc(n) local F, f;
F:= ifactors(n)[2];
mul(seq(`if`(f[1]=2, 2^f[2], floor(f[1]^(f[2]+1)/(2*f[1]+2))+1), f=F))
end proc:
map(a, [$1..100]); # Robert Israel, Jul 13 2015

f[n_] := Block[{fi = FactorInteger@ n, k = t = 1}, lng = 1 + Length@ fi; While[k < lng, t = t*If[ fi[[k, 1]] == 2, 2^fi[[k, 2]], Floor[1 + fi[[k, 1]]^(fi[[k, 2]] + 1)/(2 + 2fi[[k, 1]]) ]]; k++]; t]; Array[f, 75] (* Robert G. Wilson v, Sep 16 2015, after Robert Israel *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2^f[i,2], (f[i,1]^(f[i,2]+1)\(2*f[i,1] + 2)) + 1));} \\ Amiram Eldar, Sep 05 2023

Multiplicative with a(2^e) = 2^e, a(p^e) = floor(p^(e+1)/(2p+2))+1 for p > 2.

A122906 Numbers m such that in Z/mZ the number of squares is less than or equal to the number of invertible elements.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93

Offset: 1

Max Alekseyev, Sep 18 2006

nonn

Numbers m such that A000224(m) <= A000010(m).

The terms 70 and 90 show that sequence is strictly different from A042965. - Andrew S. Plewe, Jun 13 2007

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

Union of A122903 and A122904.
Complement of A122905 in positive integers.
Cf. A000010, A000224, A122907.

Select[Range@94, Length@Union@Mod[Range[#]^2, #] <= EulerPhi[#] &] (* Ivan Neretin, Dec 14 2016 *)
f1[p_, e_] := Floor[p^(e+1)/(2p + 2)] + 1; f1[2, e_] := Floor[2^e/6] + 2; f[p_, e_] := f1[p, e]/((p-1) * p^(e-1)); q[1] = True; q[k_] := Times @@ f @@@ FactorInteger[k] <= 1; Select[Range[100], q] (* Amiram Eldar, Nov 11 2024 *)

A186646 Every fourth term of the sequence of natural numbers 1,2,3,4,... is halved.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 4, 9, 10, 11, 6, 13, 14, 15, 8, 17, 18, 19, 10, 21, 22, 23, 12, 25, 26, 27, 14, 29, 30, 31, 16, 33, 34, 35, 18, 37, 38, 39, 20, 41, 42, 43, 22, 45, 46, 47, 24, 49, 50, 51, 26, 53, 54, 55, 28, 57, 58, 59, 30, 61, 62, 63, 32, 65, 66, 67, 34, 69, 70, 71, 36, 73, 74, 75, 38, 77, 78, 79, 40, 81, 82, 83, 42, 85, 86, 87, 44, 89, 90, 91, 46, 93, 94, 95, 48, 97, 98, 99

Offset: 1

R. J. Mathar, Feb 25 2011

a(n) is the length of the period of the sequence k^2 mod n, k=1,2,3,4,..., i.e., the length of the period of A000035 (n=2), A011655 (n=3), A000035 (n=4), A070430 (n=5), A070431 (n=6), A053879 (n=7), A070432 (n=8), A070433 (n=9), A008959 (n=10), A070434 (n=11), A070435 (n=12) etc.
From Franklin T. Adams-Watters, Feb 24 2011: (Start)
Clearly if gcd(n,m) = 1, a(nm) = lcm(a(n),a(m)), so it suffices to establish this for prime powers.
If p is a prime, the period must divide p, but k^2 mod p is not constant, so a(p) = p.
a(p^e), e > 1, must be divisible by a(p^(e-1)), and must divide p^e. If p != 2, (p^(e-1)+1)^2 =  p^(2e-2)+2p^(e-1)+1 == 2p^(e-1)+1 (mod p^2), so a(p^e) != p^(e-1); it must then be e.
By inspection, a(4) = 2 and a(8) = 4.
This leaves a(2^e), e > 3. But then (2^(e-2)+1)^2 = 2^(2e-4)+2^(e-1)+1 == 2^(e-1)+1 (mod 2^e), so a(n) > 2^(e-2). On the other hand, (2^(e-1)+c)^2 = 2^(2e-2)+c2^e+c^2 == c^2 (mod 2^e). Hence the period is 2^(e-1). (End)

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A000224 (size of the set of moduli of k^2 mod n), A019554, A060819, A061037, A090129, A142705, A164115, A283971.

A186646 := proc(n) if n mod 4 = 0 then n/2 ; else n ; end if; end proc ;

Flatten[Table[{n,n+1,n+2,(n+3)/2},{n,1,101,4}]] (* or *) LinearRecurrence[ {0,0,0,2,0,0,0,-1},{1,2,3,2,5,6,7,4},100] (* Harvey P. Dale, May 30 2014 *)
Table[n (7 - (-1)^n - 2 Cos[n Pi/2])/8, {n, 100}] (* Federico Provvedi , Jan 02 2018 *)

a(n)=if(n%4,n,n/2) \\ Charles R Greathouse IV, Oct 16 2015

def A186646(n): return n if n&3 else n>>1 # Chai Wah Wu, Jan 10 2023

a(n) = 2*a(n-4) - a(n-8).
a(4n) = 2n; a(4n+1) = 4n+1; a(4n+2) = 4n+2; a(4n+3) = 4n+3.
a(n) = n/A164115(n).
G.f.: x*(1 + 2*x + 3*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6) / ( (x-1)^2*(1+x)^2*(x^2+1)^2 ).
Dirichlet g.f.: (1-2/4^s)*zeta(s-1).
A019554(n) | a(n). - Charles R Greathouse IV, Feb 24 2011
a(n) = n*(7 - (-1)^n - (-i)^n - i^n)/8, with i=sqrt(-1). - Bruno Berselli, Feb 25 2011
Multiplicative with a(p^e)=2^e if p=2 and e<=1; a(p^e)=2^(e-1) if p=2 and e>=2; a(p^e)=p^e otherwise. - David W. Wilson, Feb 26 2011
a(n) * A060819(n+2) = A142705(n+1) = A061037(2n+2). - Paul Curtz, Mar 02 2011
a(n) = n - (n/2)*floor(((n-1) mod 4)/3). - Gary Detlefs, Apr 14 2013
a(2^n) = A090129(n+1). - R. J. Mathar, Oct 09 2014
a(n) = n*(7 - (-1)^n - 2*cos(n*Pi/2))/8. - Federico Provvedi, Jan 02 2018
E.g.f.: (1/4)*x*(4*cosh(x) + sin(x) + 3*sinh(x)). - Stefano Spezia, Jan 26 2020
Sum_{k=1..n} a(k) ~ (7/16) * n^2. - Amiram Eldar, Nov 28 2022

A290732 Number of distinct values of X(3X-1)/2 mod n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 4, 8, 9, 6, 6, 12, 7, 8, 9, 16, 9, 18, 10, 12, 12, 12, 12, 24, 11, 14, 27, 16, 15, 18, 16, 32, 18, 18, 12, 36, 19, 20, 21, 24, 21, 24, 22, 24, 27, 24, 24, 48, 22, 22, 27, 28, 27, 54, 18, 32, 30, 30, 30, 36

Offset: 1

N. J. A. Sloane, Aug 10 2017

			The values taken by (3*X^2-X)/2 mod n for small n are:
   1, [0]
   2, [0, 1]
   3, [0, 1, 2]
   4, [0, 1, 2, 3]
   5, [0, 1, 2]
   6, [0, 1, 2, 3, 4, 5]
   7, [0, 1, 2, 5]
   8, [0, 1, 2, 3, 4, 5, 6, 7]
   9, [0, 1, 2, 3, 4, 5, 6, 7, 8]
  10, [0, 1, 2, 5, 6, 7]
  11, [0, 1, 2, 4, 5, 7]
  12, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], (24-August-2016). See Table 6.

Cf. A000224 (analog for X^2), A014113, A290729, A290730, A290731, A317623.

a:=[]; M:=80;
for n from 1 to M do
q1:={};
for i from 0 to 2*n-1 do q1:={op(q1), i*(3*i-1)/2 mod n}; od;
s1:=sort(convert(q1,list));
a:=[op(a),nops(s1)];
od:
a;

a[n_] := Table[PolynomialMod[X(3X-1)/2, n], {X, 0, 2*n-1}]// Union // Length;
Array[a, 60] (* Jean-François Alcover, Sep 01 2018 *)

a(n)={my(v=vector(n)); for(i=0, 2*n-1, v[i*(3*i-1)/2%n + 1]=1); vecsum(v)} \\ Andrew Howroyd, Oct 27 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); if(p<=3, p^e, 1 + p^(e+1)\(2*p+2)))} \\ Andrew Howroyd, Nov 03 2018

a(3^n) = 3^n. - Hugo Pfoertner, Aug 25 2018
a(n) = A317623(n) * A040001(n). - Andrew Howroyd, Oct 27 2018
Multiplicative with a(2^e) = 2^e, a(3^e) = 3^e, a(p^e) = 1 + floor( p^(e+1)/(2*p+2) ) for prime p >= 5. - Andrew Howroyd, Nov 03 2018

Even terms corrected by Andrew Howroyd, Nov 03 2018

A122904 Numbers m such that in Z/mZ the number of squares is strictly less than the number of invertible elements.

Original entry on oeis.org

5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96, 97, 99, 100

Offset: 1

Max Alekseyev, Sep 18 2006

nonn

Numbers m such that A000224(m) < A000010(m).

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

Cf. A000010, A000224, A122903, A122905, A122906, A122907.

Select[Range@100, Length@Union@Mod[Range[#]^2, #] < EulerPhi[#] &] (* Ivan Neretin, Dec 14 2016 *)
f1[p_, e_] := Floor[p^(e+1)/(2p + 2)] + 1; f1[2, e_] := Floor[2^e/6] + 2; f[p_, e_] := f1[p, e]/((p-1) * p^(e-1)); q[1] = False; q[k_] := Times @@ f @@@ FactorInteger[k] < 1; Select[Range[100], q] (* Amiram Eldar, Nov 11 2024 *)

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A095972 Number of quadratic nonresidues modulo n.

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A039300 Number of distinct quadratic residues mod 3^n.

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A085635 Compute S, the number of different quadratic residues modulo B for every base B. If the density S/B is smaller for B than for every B' less than B, then B is added to the sequence.

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A118382 Primitive Orloj clock sequences; row n sums to 2n-1.

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A084848 a(n) is the number of quadratic residues of A085635(n).

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A117484 Number of triangular numbers mod n.

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A290732 Number of distinct values of X(3X-1)/2 mod n.