A138191 -id:A138191 - cp's OEIS frontend

A177704 Period 4: repeat [1, 1, 1, 2].

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 0

Klaus Brockhaus, May 11 2010

Continued fraction expansion of (3 + 2*sqrt(6))/5.
Decimal expansion of 1112/9999.
a(n) = A164115(n + 1) = (-1)^(n + 1) * A164117(n + 1) = A138191(n + 3) = A107453(n + 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David Nacin, Van der Laan Sequences and a Conjecture on Padovan Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001068, A107453, A138191, A164115, A164117, A177705, A219977, A236398.

Magma
```
&cat[ [1, 1, 1, 2]: k in [1..27] ];
```

A177704:=n->floor((n+1)*5/4) - floor(n*5/4): seq(A177704(n), n=0..100); # Wesley Ivan Hurt, Jun 15 2016

Table[Floor[(n + 1)*5/4] - Floor[n*5/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 15 2016 *)
LinearRecurrence[{0, 0, 0, 1}, {1, 1, 1, 2}, 100] (* Vincenzo Librandi, Jun 16 2016 *)

a(n) = if(n%4==3, 2, 1) \\ Felix Fröhlich, Jun 15 2016

a(n) = (5-(-1)^n + i*i^n-i*(-i)^n)/4 where i = sqrt(-1).
a(n) = a(n-4) for n > 3; a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2.
G.f.: (1+x+x^2+2*x^3)/(1-x^4).
a(n) = 1 + (1-(-1)^n) * (1+i^(n+1))/4 where i = sqrt(-1). - Bruno Berselli, Apr 05 2011
a(n) = 5/4 - sin(Pi*n/2)/2 - (-1)^n/4. - R. J. Mathar, Oct 08 2011
a(n) = floor((n+1)*5/4) - floor(n*5/4). - Hailey R. Olafson, Jul 23 2014
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n+3) - a(n+2) = A219977(n).
Sum_{i=0..n-1} a(i) = A001068(n). (End)
E.g.f.: (-sin(x) + 3*sinh(x) + 2*cosh(x))/2. - Ilya Gutkovskiy, Jun 15 2016

A385195 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is either 1 or 2.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 6, 7, 8, 8, 10, 6, 12, 12, 8, 15, 16, 16, 18, 12, 12, 20, 22, 14, 24, 24, 26, 18, 28, 16, 30, 31, 20, 32, 24, 24, 36, 36, 24, 28, 40, 24, 42, 30, 32, 44, 46, 30, 48, 48, 32, 36, 52, 52, 40, 42, 36, 56, 58, 24, 60, 60, 48, 63, 48, 40, 66, 48, 44

Offset: 1

Amiram Eldar, Jun 21 2025

			For n = 6, the greatest divisor of k that is a unitary divisor of 6 for k = 1 to 6 is 1, 2, 3, 2, 1 and 6, respectively. 4 of the values are either 1 or 2, and therefore a(6) = 4.

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

The unitary analog of A126246 (with respect to the definition "the number of integers k from 1 to n such that gcd(n,k) is either 1 or 2").
Cf. A065463, A077610, A138191, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough), this sequence (1 or 2), A385196 (prime), A385197 (noncomposite), A385198 (prime power), A385199 (1 or prime power).

f[p_, e_] := p^e - 1; f[2, 1] = 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~,f[i,1]^f[i,2] - if(f[i,1] == 2 && f[i,2] == 1, 0, 1));}

Multiplicative with a(p^e) = 2 if p = 2 and e = 1, and p^e - 1 otherwise.
In general, the number of integers k from 1 to n such that ugcd(n, k), the greatest divisor of k that is a unitary divisor of n, is either 1 or a prime power q is a multiplicative function f(n) with f(p^e) = q if p^e = q, and p^e - 1 otherwise.
a(n) = A138191(n) * A047994(n), i.e., a(n) = 2*A047994(n) if n == 2 (mod 4) and A047994(n) otherwise.
In general, the number of integers k from 1 to n such that ugcd(n, k) is either 1 or a prime power q is (q/(q-1))*A047994(n) if q is a unitary divisor of n, and A047994(n) otherwise.
Sum_{k=1..n} a(k) ~ (23/40) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.
In general, the average order of the number of integers k from 1 to n such that ugcd(n, k) is either 1 or a prime p is ((p^4+p^3-1)/(p^4+p^3-p^2)) * c * n^2 / 2, where c = A065463.

A138190 Numerator of (n-1)n(n+1)/12.

Original entry on oeis.org

0, 1, 2, 5, 10, 35, 28, 42, 60, 165, 110, 143, 182, 455, 280, 340, 408, 969, 570, 665, 770, 1771, 1012, 1150, 1300, 2925, 1638, 1827, 2030, 4495, 2480, 2728, 2992, 6545, 3570, 3885, 4218, 9139, 4940, 5330, 5740, 12341, 6622, 7095, 7590, 16215, 8648

Offset: 1

Eric W. Weisstein, Mar 04 2008

			0, 1/2, 2, 5, 10, 35/2, 28, 42, 60, 165/2, 110, 143, 182, ...

Colin Barker, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Kirchhoff Index.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Cf. A000295, A138191.

Table[(n^3-n)/12,{n,50}]//Numerator (* or *) LinearRecurrence[{0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1},{0,1,2,5,10,35,28,42,60,165,110,143,182,455,280,340},50] (* Harvey P. Dale, Nov 05 2021 *)

a(n) = numerator((n-1)*n*(n+1)/12); \\ Michel Marcus, Feb 17 2015

a(n)=binomial(n+1,3)/if(n%4==2,1,2) \\ Charles R Greathouse IV, Feb 17 2015

a(n+2) = numerator of A000295(n+2)/(3*Integral_{t=0..2} t^n*(1-abs(1-t))^2).
a(n) = (n*(n-1)*(n+1)*(5+((-1)^n-(-1)^((2*n-1+(-1)^n)/4)-(-1)^((6*n-1+(-1)^n)/4))))/48. - Luce ETIENNE, Feb 17 2015
G.f.: x^2*(x^12 +2*x^11 +5*x^10 +10*x^9 +31*x^8 +20*x^7 +22*x^6 +20*x^5 +31*x^4 +10*x^3 +5*x^2 +2*x +1) / ((x -1)^4*(x +1)^4*(x^2 +1)^4). - Colin Barker, Feb 17 2015
Sum_{n>=2} 1/a(n) = 3 * (1 - log(2)/2). - Amiram Eldar, Aug 11 2022

A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Michael Somos, Aug 10 2009

The sequence A107453 has the same terms but different offset.
Convolution inverse of A164116.
Decimal expansion of 11111/99990. - Elmo R. Oliveira, Feb 18 2024

			1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + ...

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

Cf. A107453, A138191, A164116, A164117.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2+x^3+x^4)/(1-x^4))); // G. C. Greubel, Sep 22 2018

CoefficientList[Series[(1+x+x^2+x^3+x^4)/(1-x^4), {x, 0, 100}], x] (* G. C. Greubel, Sep 22 2018 *)
LinearRecurrence[{0,0,0,1},{1,1,1,1,2},120] (* or *) PadRight[{1},120,{2,1,1,1}] (* Harvey P. Dale, Aug 24 2019 *)

PARI
```
{a(n) = 2 - (n==0) - (n%4>0)}
```

x='x+O('x^99); Vec((1-x^5)/((1-x)*(1-x^4))) \\ Altug Alkan, Sep 23 2018

Euler transform of length-5 sequence [ 1, 0, 0, 1, -1].
a(n) is multiplicative with a(2) = 1, a(2^e) = 2 if e>1, a(p^e) = 1 if p>2.
a(n) = (-1)^n * A164117(n).
a(4*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1.
a(-n) = a(n). a(n+4) = a(n) unless n=0 or n=-4.
G.f.: (1 + x + x^2 + x^3 + x^4) / ((1+x)*(1-x)*(1+x^2)).
a(n) = A138191(n+2), n>0. - R. J. Mathar, Aug 17 2009
Dirichlet g.f. (1+1/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011
a(n) = (i^n + (-i)^n + (-1)^n + 5)/4 for n > 0 where i is the imaginary unit. - Bruno Berselli, Feb 25 2011

A236398 Period 4: repeat 1,1,2,1.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jan 29 2014

H. Blaine Lawson, Jr. and M.-L. Michelsohn, Spin Geometry, Princeton, p. 33.

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

Cf. A177704, A138191.

&cat [[1, 1, 2, 1]^^24]; // Bruno Berselli, Dec 10 2015

a(n)=n=n%8; if(n==3||n==7, 2, 1) \\ Charles R Greathouse IV, Jul 17 2016

a(n) = a(n-4) for n > 4. G.f.: x*(1 + x + 2*x^2 + x^3)/(1 - x^4). - Chai Wah Wu, Jun 04 2016
E.g.f.: (-2 - sin(x) + 3*sinh(x) + 2*cosh(x))/2. - Ilya Gutkovskiy, Jun 04 2016
a(n) = (5-cos(n*Pi)-2*sin(n*Pi/2))/4. - Luce ETIENNE, Feb 17 2017

Definition corrected by Paul Curtz. - N. J. A. Sloane, Oct 10 2016

A246943 a(4n) = 4n , a(2n+1) = 8n+4 , a(4n+2) = 2*n+1.

Original entry on oeis.org

0, 4, 1, 12, 4, 20, 3, 28, 8, 36, 5, 44, 12, 52, 7, 60, 16, 68, 9, 76, 20, 84, 11, 92, 24, 100, 13, 108, 28, 116, 15, 124, 32, 132, 17, 140, 36, 148, 19, 156, 40, 164, 21, 172, 44, 180, 23, 188, 48, 196, 25, 204, 52, 212, 27, 220, 56, 228

Offset: 0

Paul Curtz, Sep 08 2014

Consider the denominators of the Balmer series A061038(n) = 0, 4, 1, 36, 16, 100,... (a permutation of the squares of the nonnegative numbers i.e. A000290(n)) divided by A028310(n)=1,1,2,... . The numerators are a(n). The denominators are A138191(n).
Note that A061038(3n)=9*A061038(n), n>=1.
a(3n) is divisible by the period 3 sequence: repeat 9, 3, 3.

			Numerators of a(0)=0/1=0, a(1)=4/1=4, a(2)=1/2, a(3)=36/3=12,... .

J. J. Balmer, Notiz über die Spectrallinien des Wasserstoffs, Annalen der Physik, vol. 261, 5 (1885) 80-87. First published June 25 1884 (Basel).
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A145979, A138191, A061038, A028310, A022998.

A246943:=n->n*(19-(-1)^n*13+2*cos(n*Pi/2))/8: seq(A246943(n), n=0..100); # Wesley Ivan Hurt, Apr 18 2017

LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,1,12,4,20,3,28},60] (* Harvey P. Dale, Jun 22 2022 *)

concat(0, Vec(x*(4*x^6+x^5+12*x^4+4*x^3+12*x^2+x+4)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100))) \\ Colin Barker, Sep 08 2014

Numerators of A061038(n)/A028310(n).
a(2n) = A022998(n).
G.f.: x*(4*x^6+x^5+12*x^4+4*x^3+12*x^2+x+4) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, Sep 08 2014
a(n) = n*(19-13*(-1)^n+(1+(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/8. - Luce ETIENNE, May 26 2015
a(n) = n*(19-(-1)^n*13+2*cos(n*Pi/2))/8. - Giovanni Resta, May 26 2015

A371124 a(n) is the least nonnegative integer y such that y^2 = x^2 - k*n for k and x where n > k >= 1 and n > x >= floor(sqrt(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 1, 0, 4, 5, 2, 6, 6, 1, 0, 8, 0, 9, 4, 2, 10, 11, 1, 0, 12, 3, 6, 14, 2, 15, 2, 4, 16, 1, 0, 18, 18, 5, 3, 20, 4, 21, 10, 2, 22, 23, 1, 0, 0, 7, 12, 26, 6, 3, 5, 8, 28, 29, 2, 30, 30, 1, 0, 4, 8, 33, 16, 10, 2, 35, 3, 36, 36, 5, 18, 2, 10

Offset: 1

Darío Clavijo, Mar 11 2024

nonn

a(A000290(n)) = 0.
a(A077591(n)) = 0.
a(A005563(n)) = 1.
For each n: k = A138191(n) and x = A306284(n).

			 n  | k | x | y^2 = x^2 - k*n  | y
------------------------------------
 1  | 1 | 1 | 0^2 = 1^2 - 1*1  | 0
 2  | 2 | 2 | 0^2 = 2^2 - 2*1  | 0
 11 | 1 | 6 | 5^2 = 6^2 - 1*11 | 5

Cf. A000040, A000290, A005563, A077591, A102781, A138191, A306284.

from sympy.core.power import isqrt
from sympy.ntheory.primetest import is_square
def a(n):
  x = isqrt(n)
  while True:
    for y2 in range(x**2-n, -1, -n):
      if is_square(y2): return isqrt(y2)
    x+=1
print([a(n) for n in range(1, 79)])

from itertools import count
def A371124(n):
    y, a = 0, {}
    for x in count(0):
        if y in a: return a[y]
        a[y] = x
        y = (y+(x<<1)+1)%n # Chai Wah Wu, Apr 25 2024

a(n) = floor(sqrt(A306284(n)^2 - n*A138191(n))).

a(A000040(n)) = A102781(n).

cp's OEIS Frontend

A177704 Period 4: repeat [1, 1, 1, 2].

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A385195 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is either 1 or 2.

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A138190 Numerator of (n-1)*n*(n+1)/12.

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A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

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A236398 Period 4: repeat 1,1,2,1.

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A246943 a(4n) = 4*n , a(2n+1) = 8*n+4 , a(4n+2) = 2*n+1.

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A371124 a(n) is the least nonnegative integer y such that y^2 = x^2 - k*n for k and x where n > k >= 1 and n > x >= floor(sqrt(n)).

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A138190 Numerator of (n-1)n(n+1)/12.

A246943 a(4n) = 4n , a(2n+1) = 8n+4 , a(4n+2) = 2*n+1.