A193509 -id:A193509 - cp's OEIS frontend

A194508 First coordinate of the (2,3)-Lagrange pair for n.

-1, 1, 0, 2, 1, 0, 2, 1, 3, 2, 1, 3, 2, 4, 3, 2, 4, 3, 5, 4, 3, 5, 4, 6, 5, 4, 6, 5, 7, 6, 5, 7, 6, 8, 7, 6, 8, 7, 9, 8, 7, 9, 8, 10, 9, 8, 10, 9, 11, 10, 9, 11, 10, 12, 11, 10, 12, 11, 13, 12, 11, 13, 12, 14, 13, 12, 14, 13, 15, 14, 13, 15, 14, 16, 15, 14, 16, 15, 17, 16, 15, 17

Offset: 1

Clark Kimberling, Aug 27 2011

Suppose that c and d are relatively prime integers satisfying 1 < c < d. Every integer n has a representation
(1) n = c*x + d*y
where x and y are integers satisfying
(2) |x - y| < d.
Let h = (c-1)*(d-1). If n >= h, there is exactly one pair (x,y) satisfying (1) and (2), and, for this pair, x >= 0 and y >= 0.
For n >= h, write (x,y) as (x(n),y(n)) and call this the (c,d)-Lagrange pair for n. If n > c*d then
(3) x(n) = x(n-c-d) + 1 and
(4) y(n) = y(n-c-d) + 1.
If n < h, then n may have more than one representation satisfying (1) and (2); e.g., 1 = 2*(-3) + 7*1 = 2*4 + 7*(-1). To extend the definition of (c,d)-Lagrange pair by stipulating a particular pair (x(n),y(n)) satisfying (1) and (2) for n < h, we reverse (3) and (4): x(n) = x(n+c+d) - 1 and y(n) = y(n+c+d) - 1 for all integers n. The initial numbers x(1) and y(1) so determined are also the numbers found by the Euclidean algorithm for 1 as a linear combination c*x + d*y.
Examples:
  c  d      x(n)      y(n)
  -  -    -------   -------
  2  3    A194508   A194509
  2  5    A194510   A194511
  2  7    A194512   A194513
  3  4    A194514   A194515
  3  5    A194516   A194517
  3  7    A194518   A194519
  3  8    A194520   A194521
  4  5    A194522   A194523
  4  7    A194524   A194525
  5  6    A194526   A194527
  5  8    A194528   A194529

			This table shows (x(n),y(n)) for 1 <= n <= 13:
   n      1  2  3  4  5  6  7  8  9 10 11 12 13
  ----   -- -- -- -- -- -- -- -- -- -- -- -- --
  x(n)   -1  1  0  2  1  0  2  1  3  2  1  3  2
  y(n)    1  0  1  0  1  2  1  2  1  2  3  2  3

L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Chelsea, 1952, page 47.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A193509-A194529.

A0:= [-1,1,0,2,0]:
f:= n -> A0[(n-1 mod 5)+1]+floor(n/5):
map(f, [$1..100]); # Robert Israel, Jul 29 2019

c = 2; d = 3;
x1 = {-1, 1, 0, 2, 1}; y1 = {1, 0, 1, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}] (* A194508 *)
Table[y[n], {n, 1, 100}] (* A194509 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

PARI
```
a(n)=2*n - (3*n+2)\5*3
```

From Robert Israel, Jul 29 2019: (Start)
a(n+5) = a(n) + 1.
G.f.: x*(-1+2*x-x^2+2*x^3-x^4)/(1-x-x^5+x^6). (End)
a(n) = 2*n - 3*floor((3*n+2)/5). - Ridouane Oudra, Sep 06 2020
a(n) = n/5 + O(1). - Charles R Greathouse IV, Mar 30 2022

A193512 a(n) = Sum of odd divisors of Omega(n), a(1) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 4, 4, 1, 4, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 4, 1, 1, 6, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 4, 1, 4, 1, 4, 1, 6, 1, 1, 4, 4, 1, 4, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 4, 1, 1, 4, 1, 1, 4

Offset: 1

Michel Lagneau, Jul 29 2011

nonn

Omega = A001222 is the number of prime divisors of the argument, counted with multiplicity.

a(1) = 0 by convention.

			a(8) = 4 because Omega(8) = 3 and the sum of the 2 odd divisors {1, 3} is 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000593, A001222, A193509, A193511, A290080.

Table[Total[Select[Divisors[PrimeOmega[n]], OddQ[ # ]&]], {n, 58}]

A000593(n) = sigma(n>>valuation(n, 2)); \\ This function from Charles R Greathouse IV, Sep 09 2014
A193512(n) = if(1==n,0,A000593(bigomega(n))); \\ Antti Karttunen, Jul 23 2017

a(1) = 0, for n > 1, a(n) = A000593(A001222(n)).

a(n) + A193511(n) = A290080(n). - Antti Karttunen, Jul 23 2017

Description clarified, more terms from Antti Karttunen, Jul 23 2017

A193510 Number of even divisors of Omega(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Michel Lagneau, Jul 29 2011

nonn

Omega(n) is the number of prime divisors of n counted with multiplicity, A001222 (also called bigomega(n)).

Records are at 4^A002182(n). [Charles R Greathouse IV, Jul 29 2011]

			a(16) = 2 because Omega(16) = 4 and the 2 even divisors are {2, 4}.

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

Cf. A183063, A001222, A193509.

f[n_] := Block[{d = Divisors[PrimeOmega[n]]}, Count[EvenQ[d], True]]; Table[f[n], {n, 80}]
Join[{0},Table[Count[Divisors[PrimeOmega[n]],?EvenQ],{n,2,100}]] (* _Harvey P. Dale, Jul 05 2023 *)

a(n)=my(k=bigomega(n));if(!k||k%2,0,numdiv(k/2)) \\ Charles R Greathouse IV, Jul 29 2011

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A194508 First coordinate of the (2,3)-Lagrange pair for n.

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A193512 a(n) = Sum of odd divisors of Omega(n), a(1) = 0.

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A193510 Number of even divisors of Omega(n).

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