A140472 -id:A140472 - cp's OEIS frontend

A214546 First differences of A140472.

1, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 8, -7, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 16, -15, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 8, -7, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0, 32, -31, 1, 0, 2, -1, 1, 0, 4, -3, 1, 0, 2, -1, 1, 0

Offset: 0

Reinhard Zumkeller, Jul 20 2012

sign

a(n) = A140472(n+1) - A140472(n);
a(A016825(n)) = 0; a(A042965(n)) <> 0;
for n > 0: a(A008586(n)) < 0, a(A005843(n)) <= 0, a(A042968(n)) >= 0;
a(A004273(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A093879.

a214546 n = a214546_list !! n
a214546_list = zipWith (-) (tail a140472_list) a140472_list

A109168 Continued fraction expansion of the constant x (A109169) such that the continued fraction of 2*x yields the continued fraction of x interleaved with the positive even numbers.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 4, 8, 5, 6, 6, 8, 7, 8, 8, 16, 9, 10, 10, 12, 11, 12, 12, 16, 13, 14, 14, 16, 15, 16, 16, 32, 17, 18, 18, 20, 19, 20, 20, 24, 21, 22, 22, 24, 23, 24, 24, 32, 25, 26, 26, 28, 27, 28, 28, 32, 29, 30, 30, 32, 31, 32, 32, 64, 33, 34, 34, 36, 35, 36, 36, 40, 37, 38, 38

Offset: 1

Paul D. Hanna, Jun 21 2005

Compare with continued fraction A100338.

The sequence is equal to the sequence of positive integers (1, 2, 3, 4, ...) interleaved with the sequence multiplied by two, 2*(1, 2, 2, 4, 3, ...) = (2, 4, 4, 8, 6, ...): see the first formula. - M. F. Hasler, Oct 19 2019

			x=1.408494279228906985748474279080697991613998955782051281466263817524862977...
The continued fraction expansion of 2*x = A109170:
[2;1, 4,2, 6,2, 8,4, 10,3, 12,4, 14,4, 16,8, 18,5, ...]
which equals the continued fraction of x interleaved with the even numbers.

Cf. A109169 (digits of x), A109170 (continued fraction of 2*x), A109171 (digits of 2*x).
Cf. A006519 and A129760. [Johannes W. Meijer, Jun 22 2011]
Half the terms of A285326.

nmax:=75; pmax:= ceil(log(nmax)/log(2)); for p from 0 to pmax do for n from 1 to nmax do a((2*n-1)*2^p):= n*2^p: od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jun 22 2011

PARI
```
a(n)=if(n%2==1,(n+1)/2,2*a(n/2))
```

A109168(n)=(n+bitand(n,-n))\2 \\ M. F. Hasler, Oct 19 2019

;; With memoization-macro definec
(definec (A109168 n) (if (zero? n) n (if (odd? n) (/ (+ 1 n) 2) (* 2 (A109168 (/ n 2))))))
;; Antti Karttunen, Apr 19 2017

a(2*n-1) = n, a(2*n) = 2*a(n) for all n >= 1.
a((2*n-1)*2^p) = n * 2^p, p >= 0. - Johannes W. Meijer, Jun 22 2011
a(n) = n - (n AND n-1)/2. - Gary Detlefs, Jul 10 2014
a(n) = A285326(n)/2. - Antti Karttunen, Apr 19 2017
a(n) = A140472(n). - M. F. Hasler, Oct 19 2019

A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

Original entry on oeis.org

1, 2, 5, 4, 8, 10, 11, 8, 20, 16, 17, 20, 20, 22, 42, 16, 26, 40, 29, 32, 58, 34, 35, 40, 53, 40, 74, 44, 44, 84, 47, 32, 90, 52, 94, 80, 56, 58, 106, 64, 62, 116, 65, 68, 174, 70, 71, 80, 102, 106, 138, 80, 80, 148, 146, 88, 154, 88, 89, 168, 92, 94, 241, 64, 172

Offset: 1

Ilya Gutkovskiy, Oct 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A328203, Sequence Machine.

Cf. A000005, A000079 (fixed points), A000203, A001227, A002131, A003602, A006519, A006918, A026741, A038040, A109168, A113415, A140472, A152211, A193356, A245579, A349353 (Dirichlet inverse), A349354.

Cf. also A347957.

a:=[]; for k in [1..65] do if IsOdd(k) then a[k]:=(k * #Divisors(k) + DivisorSigma(1,k)) / 2; else a[k]:=(k * (#Divisors(k) - #Divisors(k div 2)) + DivisorSigma(1,k) - DivisorSigma(1,k div 2)) / 2;  end if; end for; a; // Marius A. Burtea, Oct 07 2019

nmax = 65; CoefficientList[Series[Sum[k x^k/(1 - x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, (n Mod[#, 2] + Boole[OddQ[n/#]] #)/2 &]; Table[a[n], {n, 1, 65}]

A328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2)); \\ Antti Karttunen, Nov 13 2021

a(n) = (n * d(n) + sigma(n)) / 2 if n odd, (n * (d(n) - d(n/2)) + sigma(n) - sigma(n/2)) / 2 if n even.
a(n) = (n * A001227(n) + A002131(n)) / 2.
a(2*n) = 2 * a(n).
From Antti Karttunen, Nov 13 2021: (Start)
The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. Both are easy to prove:
a(n) = Sum_{d|n} A003602(d) * A026741(n/d).
a(n) = Sum_{d|n} A109168(d) * A193356(n/d), where A109168(d) = A140472(d) = (d+A006519(d))/2.
(End)

A349344 Dirichlet inverse of A109168, where A109168(n) = (n+A006519(n))/2, and A006519 is the highest power of 2 dividing n.

Original entry on oeis.org

1, -2, -2, 0, -3, 4, -4, 0, -1, 6, -6, 0, -7, 8, 4, 0, -9, 2, -10, 0, 5, 12, -12, 0, -4, 14, -2, 0, -15, -8, -16, 0, 7, 18, 6, 0, -19, 20, 8, 0, -21, -10, -22, 0, 3, 24, -24, 0, -9, 8, 10, 0, -27, 4, 8, 0, 11, 30, -30, 0, -31, 32, 4, 0, 9, -14, -34, 0, 13, -12, -36, 0, -37, 38, 8, 0, 9, -16, -40, 0, -4, 42, -42, 0

Offset: 1

Antti Karttunen, Nov 15 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A109168 (A140472), A349345.

Cf. also A328203, A349134, A349343, A349346, A349353.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA109168(n) = ((n+bitand(n, -n))\2); \\ From A109168 by M. F. Hasler, Oct 19 2019 (Cf. A140472).
v349344 = DirInverseCorrect(vector(up_to,n,A109168(n)));
A349344(n) = v349344[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A109168(n/d) * a(d).

a(n) = A349345(n) - A109168(n).

A285326 a(0) = 0, for n > 0, a(n) = n + A006519(n).

Original entry on oeis.org

0, 2, 4, 4, 8, 6, 8, 8, 16, 10, 12, 12, 16, 14, 16, 16, 32, 18, 20, 20, 24, 22, 24, 24, 32, 26, 28, 28, 32, 30, 32, 32, 64, 34, 36, 36, 40, 38, 40, 40, 48, 42, 44, 44, 48, 46, 48, 48, 64, 50, 52, 52, 56, 54, 56, 56, 64, 58, 60, 60, 64, 62, 64, 64, 128, 66, 68, 68, 72, 70, 72, 72, 80, 74, 76, 76, 80, 78, 80, 80, 96, 82, 84, 84, 88, 86, 88, 88, 96, 90, 92, 92

Offset: 0

Antti Karttunen, Apr 19 2017

From M. F. Hasler, Oct 19 2019: (Start)
This sequence is equal to itself multiplied by 2 and interleaved with the positive even numbers: We have a(2n-1) = 2n (n >= 1) from the very definition, since A006519(m) = 1 for odd m. And a(2n) = 2n + A006519(2n) = 2*a(n), using A006519(2n) = 2*A006519(n).
The sequence repeats the pattern [A, B, C, C] where in the n-th occurrence C = 4n, B = C - 2, A = C if n is even, A = C + 4 if n = 3 (mod 4), and A = 16*a((n-1)/4) otherwise. (End)

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Row 2 of A285325 (after the initial zero).
Cf. A006519, A019565, A048675, A065642, A129760, A285109.
Cf. A109168 (same terms divided by 2), also A140472.

Table[If[n>0, n + GCD[2^n, n], 0], {n, 0, 100}] (* Indranil Ghosh, Apr 20 2017 *)

a(n) = if(n>0, n + gcd(2^n, n), 0); \\ Indranil Ghosh, Apr 20 2017

A285326(n)=n+bitand(n,-n) \\ Or: {a(n)=-bitand(-n,bitneg(n))}, not faster. - M. F. Hasler, Oct 19 2019

from sympy import gcd
def a(n): return n + gcd(2**n, n) if n>0 else 0 # Indranil Ghosh, Apr 20 2017

(define (A285326 n) (if (zero? n) n (+ n (A006519 n))))
(define (A285326 n) (A048675 (A065642 (A019565 n))))

a(0) = 0; for n > 0, a(n) = n + A006519(n).
a(n) = A048675(A065642(A019565(n))) = A048675(A285109(A019565(n))).
For n >= 1, a(n) = 2*A109168(n).
a(n) = 2*A140472(n) and a(2n) = 2*a(n) and a(2^n) = 2^(n+1) for all n >= 0, a(2n-1) = 2n for all n >= 1. - M. F. Hasler, Oct 19 2019

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A214546 First differences of A140472.

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A109168 Continued fraction expansion of the constant x (A109169) such that the continued fraction of 2*x yields the continued fraction of x interleaved with the positive even numbers.

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A328203 Expansion of Sum_{k>=1} k * x^k / (1 - x^(2*k))^2.

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A349344 Dirichlet inverse of A109168, where A109168(n) = (n+A006519(n))/2, and A006519 is the highest power of 2 dividing n.

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A285326 a(0) = 0, for n > 0, a(n) = n + A006519(n).

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