A003602 -id:A003602 - cp's OEIS frontend

A110812 Fractalization of sqrt 2.

1, 1, 4, 1, 1, 4, 4, 1, 2, 1, 1, 4, 3, 4, 5, 1, 6, 2, 2, 1, 3, 1, 7, 4, 3, 3, 0, 4, 9, 5, 5, 1, 0, 6, 4, 2, 8, 2, 8, 1, 0, 3, 1, 1, 6, 7, 8, 4, 8, 3, 7, 3, 2, 0, 4, 4, 2, 9, 0, 5, 9, 5

Offset: 1

Alexandre Wajnberg, Sep 15 2005

Self-descriptive sequence: even terms are the sequence itself, odd terms are the digits of the decimal expansion of sqrt 2.

Clark Kimberling, Fractal sequences.

Cf. A002193 (sqrt 2), A003602.

Cf. A110766 (of Pi), A110779 (of e), A382130 (of phi).

a(2n)=a(n); a(2n+1)=digits of sqrt 2.

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)

A347374 Lexicographically earliest infinite sequence such that a(i) = a(j) => A331410(i) = A331410(j) and A000593(i) = A000593(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 17, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 25, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 17, 32, 9, 33, 17, 34, 5, 35, 18, 36, 10, 33, 19, 37, 3, 38, 20, 39, 11, 40, 21, 41, 6, 42, 22, 43, 12

Offset: 1

Antti Karttunen, Aug 29 2021

nonn

Restricted growth sequence transform of the ordered pair [A000593(n), A331410(n)].

For all i, j: A003602(i) = A003602(j) => a(i) = a(j) => A347249(i) = A347249(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000593, A003602, A331410, A347249.

Cf. also A335880, A336390, A336391, A336394 for similar constructions.

up_to = 65537;
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; };
A000593(n) = sigma(n>>valuation(n, 2));
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
Aux347374(n) = [A331410(n), A000593(n)];
v347374 = rgs_transform(vector(up_to, n, Aux347374(n)));
A347374(n) = v347374[n];

A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 2, 2, 5, 2, 6, 3, 0, 8, 8, 2, 9, 4, 0, 5, 11, 4, 6, 6, 4, 6, 14, 0, 15, 16, 0, 8, 0, 4, 18, 9, 0, 8, 20, 0, 21, 10, -2, 11, 23, 8, 12, 6, 0, 12, 26, 4, 0, 12, 0, 14, 29, 0, 30, 15, -3, 32, 0, 0, 33, 16, 0, 0, 35, 8, 36, 18, -4, 18, 0, 0, 39, 16, 8, 20, 41, 0, 0, 21, 0, 20, 44, -2, 0, 22, 0, 23

Offset: 1

Antti Karttunen, Nov 17 2021

sign

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

Cf. A003602, A055615, A349134, A349431 (Dirichlet inverse), A349433 (sum with it).

Cf. also A349445, A349448.

k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#] * k[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
A349134(n) = v349134[n];
A003602(n) = (1+(n>>valuation(n,2)))/2;
A055615(n) = (n*moebius(n));
A349432(n) = sumdiv(n,d,d*A349134(n/d));

A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 8, 2, 12, 7, 13, 4, 14, 8, 11, 1, 15, 9, 15, 5, 16, 10, 17, 3, 18, 11, 19, 6, 20, 8, 15, 2, 21, 12, 22, 7, 23, 13, 22, 4, 24, 14, 25, 8, 26, 11, 27, 1, 28, 15, 29, 9, 30, 15, 22, 5, 31, 16, 32, 10, 30, 17, 24, 3, 33, 18, 28, 11, 34, 19, 35, 6, 36, 20, 37, 8, 38, 15, 35, 2, 39, 21, 40, 12, 41, 22, 42, 7, 43

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the ordered pair [A206787(n), A336651(n)], or equally, of sequence b(n) = A291750(A000265(n)).
For all i, j >= 1:
  A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
  A324400(i) = A324400(j) => A351460(i) = A351460(j) => a(i) = a(j),
  a(i) = a(j) => A000593(i) = A000593(j),
  a(i) = a(j) => A347385(i) = A347385(j),
  a(i) = a(j) => A351037(i) = A351037(j) => A347240(i) = A347240(j).
From Antti Karttunen, Nov 23 2023: (Start)
Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j) and A336467(i) = A336467(j) for all i, j >= 1. In any case, a(i) = a(j) => b(i) = b(j) for all i, j >= 1 [because both A000593(n) and A336467(n) can be computed from the values of A206787(n) and A336651(n)], but whether the implication holds to the opposite direction is still open. Empirically this has been checked up to n = 2^22. See also comment in A351040.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206787, A336651.
Cf. also A000593, A003602, A291750, A291751, A324400, A336467, A347240, A347385, A351040, A351460.
Differs from A351037 for the first time at n=103, where a(103) = 42 while A351037(103) = 27.

up_to = 65537;
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; };
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351461(n) = [A206787(n), A336651(n)];
v351461 = rgs_transform(vector(up_to, n, Aux351461(n)));
A351461(n) = v351461[n];

A101279 a(1) = 1; a(2k) = a(k), a(2k+1) = k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 3, 7, 1, 8, 4, 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 1, 16, 8, 17, 4, 18, 9, 19, 2, 20, 10, 21, 5, 22, 11, 23, 1, 24, 12, 25, 6, 26, 13, 27, 3, 28, 14, 29, 7, 30, 15, 31, 1, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 38, 19, 39, 2, 40, 20, 41, 10

Offset: 1

N. J. A. Sloane, May 22 2006; definition corrected May 23 2006

nonn

From Jeremy Gardiner, Mar 22 2015: (Start)
For n > 2 write n, n-1 in binary, then align bits from the left and take contiguous matching bits as a binary number.
For example:
n    = 19 10011
n-1  = 18 10010
a(n) =  9 1001
Also arrange the positive integers as a binary tree rooted at 1 as shown:
                                     1
                                     |
                 2................../ \..................3
                 |                                       |
       4......../ \........5                   6......../ \........7
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   8       9          10      11          12       13         14       15
16 17   18 19      20  21  22  23      24  25   26  27     28  29   30  31
Each branch doubles the number above at the left fork or doubles and adds 1 at the right fork. Then for n > 2, a(n) is the greatest common ancestor of n and n-1, a(n) = gca(n,n-1).
(End)
From David James Sycamore, Mar 07 2023: (Start)
The following identical sequences, {b(n)} and {c(n)}, are the same as a(n+1) for n >= 1.
b(1) = 1, then reverse the conditions in Name: b(2k) = k, b(2k+1) = b(k).
c(1) = 1, then if c(n) is a first occurrence, c(n+1) = c(c(n)), else if c(n) has occurred previously, c(n+1) = n - c(n-1).
These are fractal sequences (b(2m+1) = c(2m+1), m >= 1, recovers the originals). Also {b(n)} and {c(n)} interleave A000027 with the present sequence.
(End)

			If n is a power of 2 then k=1.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A003602, A025480.

cf. A000027, A000079, A007283, A070939, A119387. - David James Sycamore, Mar 07 2023

a:=array(0..200); a[1]:=1; M:=200; for n from 2 to M do if n mod 2 = 1 then a[n]:=(n-1)/2; else a[n]:=a[n/2]; fi; od: [seq(a[n],n=1..M)];

a[1] = 1; a[n_] := a[n] = If[OddQ@n, (n - 1)/2, a[n/2]]; Array[a, 84] (* Robert G. Wilson v, May 23 2006 *)

a(n)=(n/2^valuation(n,2)-1)/2+if(n==2^valuation(n,2),1,0) /* Ralf Stephan, Aug 21 2013 */

a((n+1)/2) = A028310(n) if n is odd and a(n/2) = a(n) if n is even; thus this is a fractal sequence. - Robert G. Wilson v, May 23 2006; corrected by Clark Kimberling, Jul 07 2007
a(n) = A025480(n) + A036987(n) = (n/2^A007814(n) - 1)/2 + (n == 2^A007814(n)). - Ralf Stephan, Aug 21 2013
If n is a power of 2, A070939(a(n)) = 1, otherwise A070939(a(n)) = A119387(n-1).
Numbers m for which a(m) = 1 are A000079(m) and A007283(m), a(2^m + 1) = 2^(m-1); m >= 1. - David James Sycamore, Mar 07 2023

A110963 Fractalization of Kimberling's paraphrases sequence beginning with 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 4, 1, 1, 1, 5, 3, 3, 2, 6, 2, 2, 1, 7, 4, 4, 1, 8, 1, 1, 1, 9, 5, 5, 3, 10, 3, 3, 2, 11, 6, 6, 2, 12, 2, 2, 1, 13, 7, 7, 4, 14, 4, 4, 1, 15, 8, 8, 1, 16, 1, 1, 1, 17, 9, 9, 5, 18, 5, 5, 3, 19, 10, 10, 3, 20, 3, 3, 2, 21, 11, 11, 6, 22, 6, 6, 2, 23, 12, 12, 2, 24, 2, 2, 1, 25, 13

Offset: 1

Alexandre Wajnberg, Sep 26 2005

Self-descriptive sequence: terms at even indices are the sequence itself, terms at odd indices (the skeleton of this sequence) are the terms of Kimberling's paraphrases sequence (A003602) beginning with 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Clark Kimberling, Fractal sequences.
Index entries for sequences related to binary expansion of n

One more than A110962 (but note the different starting offsets).
Cf. A000265, A003602, A005408, A089309, A110812, A110779, A110766, A351565 [= 2*a(n) - 1].
Cf. A353366 (Dirichlet inverse), A353367 (sum with it).

A003602(n) = (1+(n>>valuation(n,2)))/2;
A110963(n) = if(n%2, A003602((1+n)/2), A110963(n/2)); \\ Antti Karttunen, Apr 03 2022

a(n) = n>>=valuation(n,2); 1+n>>valuation(2*n+2,2); \\ Ruud H.G. van Tol, Jun 23 2024

def A110963(n): return (1+(m:=n>>(~n&n-1).bit_length())>>(m+1&-m-1).bit_length())+1 # Chai Wah Wu, Jan 04 2024

For even n, a(n) = a(n/2), for odd n, a(n) = A003602((1+n)/2). - Antti Karttunen, Apr 03 2022
For n >= 0, (Start)
a(4n+2) = a(4n+3) = A003602(1+n).
a(8n+1) = A005408(n) = 2*n + 1.
a(4n+1) = a(8n+2) = a(8n+3) = 1+n.
a(n) = A110962(n-1) + 1.
(End)
a(n) = A353367(4*n). - Antti Karttunen, Apr 20 2022
a(n) = A003602(A003602(n)). - Ridouane Oudra, Dec 28 2024

Entry edited, starting offset corrected (from 0 to 1), and the offsets in formulas changed accordingly, and more terms added by Antti Karttunen, Apr 03 2022

A264646 A simple self-describing sequence S: n concatenated with the n-th digit of S.

Original entry on oeis.org

11, 21, 32, 41, 53, 62, 74, 81, 95, 103, 116, 122, 137, 144, 158, 161, 179, 185, 191, 200, 213, 221, 231, 246, 251, 262, 272, 281, 293, 307, 311, 324, 334, 341, 355, 368, 371, 386, 391, 401, 417, 429, 431, 448, 455, 461, 479, 481, 492, 500, 510, 522, 531

Offset: 1

Eric Angelini and Reinhard Zumkeller, Nov 20 2015

Although A003602 and this sequence initially agree in their digit-streams, they differ after 48 digits. - N. J. A. Sloane, Nov 20 2015

			.   n |  1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10  | 11  | 12  | 13  | 14
. ----+----+---+---+---+---+---+---+---+---+-----+-----+-----+-----+-----
. a(n)| 11  21  32  41  53  62  74  81  95  103   116   122   137   144
. digs| 1 1 2 1 3 2 4 1 5 3 6 2 7 4 8 1 9 5 1 0 3 1 1 6 1 2 2 1 3 7 1 4 4 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, n concatenated with the nth digit of S, SeqFan list, Nov 19 2015.

Cf. A003602.

import Data.List (genericIndex)
a264646 n = a264646_list !! (n-1)
a264646_list = 11 : f 2 [0, 1, 1] where
   f x digs = (foldl (\v d -> 10 * v + d) 0 ys) : f (x + 1) (digs ++ ys)
     where ys = map (read . return) (show x) ++ [genericIndex digs x]

from itertools import count, islice
def agen(): # generator of terms
    an, s = 11, [None, 1, 1]
    for n in count(2):
        yield an
        an = 10*n + s[n]
        s.extend(list(map(int, str(an))))
print(list(islice(agen(), 53))) # Michael S. Branicky, Oct 03 2024

A336460 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A336158(n), A336466(n)], for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 3, 5, 9, 3, 10, 6, 11, 2, 12, 7, 13, 4, 14, 8, 15, 1, 16, 3, 17, 5, 18, 9, 19, 3, 20, 10, 21, 6, 22, 11, 23, 2, 24, 12, 25, 7, 26, 13, 27, 4, 28, 14, 29, 8, 30, 15, 31, 1, 32, 16, 33, 3, 34, 17, 35, 5, 18, 18, 22, 9, 36, 19, 37, 3, 38, 20, 39, 10, 40, 21, 41, 6, 42, 22, 43, 11, 44, 23, 45, 2, 7, 24, 46, 12, 47, 25, 48, 7, 49

Offset: 1

Antti Karttunen, Jul 24 2020

nonn

Restricted growth sequence transform of the ordered triple [A278222(n), A336158(n), A336466(n)].
For all i, j:
  A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j),
  a(i) = a(j) => A336159(i) = A336159(j),
  a(i) = a(j) => A336470(i) = A336470(j) => A336471(i) = A336471(j),
  a(i) = a(j) => A336472(i) = A336472(j),
  a(i) = a(j) => A336473(i) = A336473(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A278222, A336158, A336466.

Cf. A003602, A324400, A336159, A336470, A336471, A336472, A336473.

up_to = 65537;
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; };
A000265(n) = (n>>valuation(n,2));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
A336158(n) = A046523(A000265(n));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
Aux336460(n) = [A278222(n), A336158(n), A336466(n)];
v336460 = rgs_transform(vector(up_to, n, Aux336460(n)));
A336460(n) = v336460[n];

A351090 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351091(i) = A351091(j) and A351092(i) = A351092(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42, 11, 43, 22, 44, 6, 45, 23

Offset: 1

Antti Karttunen, Jan 31 2022

Restricted growth sequence transform of the ordered pair [A351091(n), A351092(n)], or equally, of the ordered pair [A351093(n), A351094(n)].

For all i, j: A003602(i) = A003602(j) => a(i) = a(j) => A000593(i) = A000593(j).

			Consider two odd semiprimes, 689 and 697. The divisors of 689 are 1, 13, 53, 689, and the divisors of 697 are 1, 17, 41, 697. Applying A019565(A289813(x)) to the former gives [2, 30, 7, 105], while with the latter it gives [2, 5, 105, 42], and the product of both sequences is 44100. Applying A019565(A289814(x)) to the former gives [1, 1, 30, 286], while with the latter it gives [1, 6, 2, 715]. Product of both sequences is 8580. Therefore, because A351091(689) = A351091(697) and A351092(689) = A351092(697), also a(689) = a(697).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000593, A351037, A351091, A351092, A351093, A351094.
Differs from A003602 for the first time at n=697, where a(697) = 345 while A003602(697) = 349.
Cf. also A293226, A351030.

up_to = 65537;
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; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); }; \\ From A289813
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); }; \\ From A289814
A351091(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289813(d))); (m); };
A351092(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289814(d))); (m); };
Aux351090(n) = [A351091(n),A351092(n)];
v351090 = rgs_transform(vector(up_to, n, Aux351090(n)));
A351090(n) = v351090[n];

cp's OEIS Frontend

A110812 Fractalization of sqrt 2.

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

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Magma

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A347374 Lexicographically earliest infinite sequence such that a(i) = a(j) => A331410(i) = A331410(j) and A000593(i) = A000593(j), for all i, j >= 1.

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A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

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A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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A101279 a(1) = 1; a(2k) = a(k), a(2k+1) = k.

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Maple

Mathematica

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A110963 Fractalization of Kimberling's paraphrases sequence beginning with 1.

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PARI

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A264646 A simple self-describing sequence S: n concatenated with the n-th digit of S.

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