A003602 -id:A003602 - cp's OEIS frontend

A365393 Lexicographically earliest infinite sequence such that a(i) = a(j) => A364492(i) = A364492(j) for all i, j >= 0, where A364492(n) is the denominator of n / A163511(n).

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

Offset: 0

Antti Karttunen, Sep 06 2023

Restricted growth sequence transform of A364492.

Question: Which sets of numbers cause the finite branches that grow off-angle from the rays emanating from the origin in the scatter plot, and why the sudden bends in some of them? Compare also to the scatter plot of A365431.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000265, A003602, A163511, A364255, A364492.

Cf. also A365384, A365431.

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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A364492(n) = { my(u=A163511(n)); (u/gcd(n, u)); };
v365393 = rgs_transform(vector(1+up_to,n,A364492(n-1)));
A365393(n) = v365393[1+n];

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

A382130 Fractalization of the golden ratio.

Original entry on oeis.org

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

Offset: 1

David Cleaver, Mar 16 2025

Self-descriptive sequence: even indexed terms are the sequence itself, odd indexed terms are the decimal digits of the golden ratio.
This is an r1k1 fractal sequence, where r1k1 means: remove 1 term, keep 1 term, repeat.  The Removed terms are the sequence that has been fractalized, and the Kept terms are the original fractal sequence.
This fractal sequence is not a Kimberling fractal sequence because if you delete the first occurrence of each term, the remaining sequence is not the same as the original.

David Cleaver, Table of n, a(n) for n = 1..10000
Clark Kimberling, Fractal sequences.

Bisection gives A001622 (odd part).

Cf. A003602, A110766, A110779, A110812, A382128, A382129.

a(2n) = a(n); a(2n-1) = A001622(n), n >= 1.

a(n) = A001622(A003602(n)).

A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 8, 1, 16, 4, 32, 2, 64, 8, 128, 1, 256, 16, 512, 4, 1024, 32, 2048, 2, 4096, 64, 8192, 8, 16384, 128, 32768, 1, 65536, 256, 131072, 16, 262144, 512, 524288, 4, 1048576, 1024, 2097152, 32, 4194304, 2048, 8388608, 2, 16777216

Offset: 0

Ralf Stephan, Jun 07 2003

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A045654, A220466.

nmax := 48: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := 2^n od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 11 2013
A082392 := proc(n)
    2^A025480(n) ;
end proc:
seq(A082392(n),n=0..100) ; # R. J. Mathar, Jul 16 2020

a[n_] := 2^(((n+1)/2^IntegerExponent[n+1, 2]+1)/2-1);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 15 2023 *)

for(n=0, 50, l=ceil(log(n+1)/log(2)); t=polcoeff(sum(k=0, l, (x^2^k)/(1-2*x^2^(k+1)))/x + O(x^(n+1)), n); print1(t", ");) ;

a(0) = 1, a(2*n) = 2^n, a(2*n+1) = a(n).
a(n) = 2^A025480(n) = 2^(A003602(n)-1).
a((2*n+1)*2^p-1) = 2^n, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 11 2013

A108202 a(2n) = A007376(n); a(2n+1) = a(n).

Original entry on oeis.org

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

Offset: 0

Juliette Bruyndonckx and Alexandre Wajnberg, Jun 15 2005

A sequence, S, consisting of the natural counting digits (A007376) interleaved with S.
Fractal sequence (Kimberling-type) based upon the counting digits.
Similar to but different from A025480 - N. J. A. Sloane, Nov 26 2020.

First differs from A025480 at a(20).
Even bisection: A007376.
Cf. A003602.

a(n) = A007376(A025480(n)). - Kevin Ryde, Nov 21 2020

More terms from Joshua Zucker, May 18 2006

Name made more explicit by Peter Munn, Nov 20 2020

A131987 Representation of a dense para-sequence.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 1, 6, 3, 7, 8, 4, 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 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, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 38, 19, 39, 2, 40, 20, 41, 10, 42, 21, 43

Offset: 1

Clark Kimberling, Aug 05 2007, Sep 12 2007

A fractal sequence. The para-sequence may be regarded as a sort of "limit" of the concatenated segments. The para-sequence (itself not a sequence) is dense in the sense that every pair of terms i and j are separated by another term (and hence separated by infinitely many terms).
The para-sequence accounts for positions of dyadic rational numbers in the following way: Label 1/2 as 1; label 1/4, 3/4 as 2 and 3; label 1/8, 3/8, 5/8, 7/8 as 4,5,6,7, etc. Then, for example, the ordering 1/8 < 1/4 < 3/8 < 1/2 < 5/8 < 3/4 < 7/8 matches the labels 4,2,5,1,6,3,7, which is the 3rd segment of A131987. The n-th segment consists of labels for rationals having denominators 2, 4, 8, ..., 2^n.
Could be seen as a "fuzzy table" with row lengths 2^n-1. In row n one has the numbers, read from the leftmost to the rightmost, as they appear in a perfect binary tree of 2^n-1 nodes when inserted in "storage order" into the tree, cf. illustration in A101279 and stackexchange link. These rows are obviously permutations of [1..2^n-1], their inverse is given in A269752. - M. F. Hasler, Mar 04 2016
Subsequence of A025480 (omitting all terms=0). - David James Sycamore, Apr 26 2020
The sequence obtained by adding 1 to every term of this sequence is the same as A003602 with all 1's removed. - David James Sycamore, Jul 25 2022

			Start with 1 and isolate it using 2,3, like this: 2,1,3.  Then isolate those using 4,5,6,7, like this: 4,2,5,1,6,3,7.  The next segment, to be concatenated after 4,2,5,1,6,3,7, is 8,4,9,2,10,5,11,1,12,6,13,3,14,7,15.

C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.

Clark Kimberling, Table of n, a(n) for n = 1..10000
Josef Eschgfäller, Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.
N.A., References to these functions relating to binary trees and binary digit counting?, Stackexchange forum, Feb 28 2016.
Clark Kimberling, Self-Containing Sequences, Selection Functions, and Parasequences, J. Int. Seq. Vol. 25 (2022), Article 22.2.1.

Cf. A001511, A025480, A070941, A133108, A003602.

m:=p->padic[ordp](2*p,2)-1:podd:=(h,p)->2^h+(p-2)/2:peven:=(h,p)->2^(h-m(p))+(p-2^m(p))/2^(m(p)+1):for i from 0 to 5 do for j from 1 to 2^(i+1)-1 do if j mod 2 =1 then print(podd(i,j)) else print(peven(i,j)) fi od od # Gary Detlefs, Sep 28 2018

Flatten@NestList[Riffle[Range[Length[#] + 1, 2 Length[#] + 1], #] &, {1}, 4] (* Birkas Gyorgy, Mar 11 2011 *)

A131987_row(n,r=[1])={for(k=2,n,r=vector(2^k-1,j,if(bittest(j,0),j\2+2^(k-1),r[j\2])));r}
apply(A131987_row,[1..6]) \\ or concat(...) \\ M. F. Hasler, Mar 04 2016

When viewed as a table, T(h,p), related to the in order traversal of a full binary tree, T(h,p) = 2^h+(p-1)/2, p odd, 2^(h-m(p)) + (p-2^m(p)) / 2^(m(p)+1), where m(p) is the greatest value of n such that p mod 2^n == 0. m(p) = p-adic[ordp](2*p,2)-1. - Gary Detlefs, Sep 28 2018

a((2*n+1)*2^k - k - A070941(n)) = n = A025480((2*n+1)*2^k - 1); (n>=1, k>=0). - David James Sycamore, Apr 26 2020

A163491 A fractal sequence (if we delete the first occurrence of n we get the sequence itself).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 4, 3, 1, 5, 2, 4, 6, 3, 1, 7, 5, 2, 8, 4, 6, 9, 3, 1, 10, 7, 5, 11, 2, 8, 12, 4, 6, 13, 9, 3, 14, 1, 10, 15, 7, 5, 16, 11, 2, 17, 8, 12, 18, 4, 6, 19, 13, 9, 20, 3, 14, 21, 1, 10, 22, 15, 7, 23, 5, 16, 24, 11, 2, 25, 17, 8, 26, 12

Offset: 1

Benoit Cloitre, Jul 29 2009

nonn

Write the positive integers with two spaces between each integer: 1,,,2,,,3,,,4,,,5,,,6,..., and fill undefined places with the sequence itself. A003602 is obtained by starting from 1,,2,,3,,4,,5,_,6,....
From Peter Munn, Aug 02 2020: (Start)
a(n) - 1 is the row of A083044 in which n occurs.
The m-th occurrence of m is at position A083045(m-1).
(End)

			  1,_,_,2,_,_,3,_,_,4,... -->
  1,1,_,2,_,_,3,_,_,4,... -->
  1,1,1,2,_,_,3,_,_,4,... -->
  1,1,1,2,1,_,3,_,_,4,... -->
  1,1,1,2,1,2,3,_,_,4,... -->
  1,1,1,2,1,2,3,_,_,4,... -->
  1,1,1,2,1,2,3,1,_,4,... -->
  1,1,1,2,1,2,3,1,2,4,... -->
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A003602, A007494, A061419, A083044, A083045.

Ordinal transform of A087088.

a[n_] := a[n] = If[Mod[n, 3] == 1, (n+2)/3, a[Floor[2n/3]]];
Array[a, 100] (* Jean-François Alcover, Jan 10 2022 *)

a(n) = n+=2; my(q,r); while([q,r]=divrem(n,3); r, n-=q); q; \\ Kevin Ryde, Jan 16 2021

def a(n): return (n+2)//3 if n%3==1 else a(n*2//3)
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Jan 16 2021

a(3n-2) = n.
A083044(a(n) - 1, A087088(n) - 1) = n. - Peter Munn, Aug 02 2020
From Rémy Sigrist, Jan 15 2021: (Start)
a(n+ceiling(n/2)) = a(n).
a(n) = 1 iff n belongs to A061419.
(End)
a(n) = (n+2)/3 if n == 1 (mod 3), otherwise a(n) = a(floor(n*2/3)). - Michael S. Branicky and Kevin Ryde, Jan 16 2021

Terms after a(70) corrected by Jon E. Schoenfield, Nov 26 2015

A265104 a(n) = A265100(n+1) - 6, n >= 1.

Original entry on oeis.org

8, 26, 35, 80, 89, 107, 116, 242, 251, 269, 278, 323, 332, 350, 359, 728, 737, 755, 764, 809, 818, 836, 845, 971, 980, 998, 1007, 1052, 1061, 1079, 1088, 2186, 2195, 2213, 2222, 2267, 2276, 2294, 2303, 2429, 2438, 2456, 2465, 2510, 2519, 2537

Offset: 1

L. Edson Jeffery, Dec 01 2015

nonn

In the following, let "gap" and "gap number" be as defined in A265100, and let C(m) denote the m-th Catalan number (A000108).
Conjecture 1: The sequence contains all possible gap numbers.
Conjecture 2: For any gap G, the order |G| of G is the constant |G| = 6.
Conjecture 3: If g is a gap number, then 3*g + 2 is a gap number.
Conjecture 4: If C(m) =!= 0 (mod 3), then C(3*m+1) =!= 0 (mod 3) (=!= means "not congruent") or, what is the same thing, if m lies in a gap, then 3*m + 1 lies in a gap.

Cf. A265100.

a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265100[n_] := 9*a005836[n] + 5; a265104[n_] := a265100[n+1] - 6; Table[a265104[n], {n, 46}]
(* Or: *)
a007814[x_] := IntegerExponent[x, 2]; a003602[x_] := (1 + x/2^a007814[x])/2; a005836[1] := 0; a005836[n_] := If[OddQ[n], 3*a005836[Floor[(n + 1)/2]], a005836[n - 1] + 1]; a265100[n_] := 9*a005836[n] + 5; a265104[n_] := (3^(a007814[n] + 2) - 3)/2 + a265100[2^(a007814[n])*(2*a003602[n] - 1)]; Table[a265104[n], {n, 46}]

a(n) = (3^(A007814(n) + 2) - 3)/2 + A265100(2^(A007814(n))*(2*A003602(n) - 1)), n >= 1.

Conjecture: a(n) = A265100(n) + A085296(n).

A336475 Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (e+1)*p^e.

Original entry on oeis.org

1, 1, 6, 1, 10, 6, 14, 1, 27, 10, 22, 6, 26, 14, 60, 1, 34, 27, 38, 10, 84, 22, 46, 6, 75, 26, 108, 14, 58, 60, 62, 1, 132, 34, 140, 27, 74, 38, 156, 10, 82, 84, 86, 22, 270, 46, 94, 6, 147, 75, 204, 26, 106, 108, 220, 14, 228, 58, 118, 60, 122, 62, 378, 1, 260, 132, 134, 34, 276, 140, 142, 27, 146, 74, 450, 38, 308, 156

Offset: 1

Antti Karttunen, Jul 30 2020

Dirichlet convolution of A000265 with itself, divided by A001511(n).

Although for all i, j: A003602(i) = A003602(j) => a(i) = a(j), it is not true that a(i) = a(j) => A003602(i) = A003602(j), because A038040 has a duplicate occurrence of a term on at least two odd positions: A038040(1875) = A038040(3125) = 18750.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A000265, A001227, A001511, A001620, A003602, A038040, A336476.

f[p_, e_] := (e+1)*p^e; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)

A336475(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,(1+f[i,2]) * (f[i,1]^f[i,2]))); };

from sympy import divisor_count
def A336475(n): return (m:=n>>(~n&n-1).bit_length())*divisor_count(m) # Chai Wah Wu, Jul 13 2022

a(n) = A038040(A000265(n)).
a(n) = A000265(n) * A001227(n).
a(n) = (A000005(n) * A000265(n)) / A001511(n). [See the first comment]
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: ((2^s - 2)^2/(4^s - 2^s)) * zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (n^2/12) * (2*log(n) + 4*gamma + 10*log(2)/3 - 1), where gamma is Euler's constant (A001620). (End)

A336935 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A278222(i) = A278222(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, 7, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 27, 4, 28, 15, 29, 8, 30, 16, 31, 1, 32, 17, 33, 9, 34, 7, 35, 5, 36, 18, 37, 10, 38, 19, 39, 3, 40, 20, 41, 11, 42, 21, 43, 6, 44, 22, 45, 12, 46, 23, 47, 2, 48, 24, 49, 13, 50

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the ordered pair [A007733(n), A278222(n)].

For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

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

Cf. A000265, A003602, A007733, A278222, A286622, A324400, A336933, A336934.

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; };
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
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));
Aux336935(n) = [A007733(n), A278222(n)];
v336935 = rgs_transform(vector(up_to, n, Aux336935(n)));
A336935(n) = v336935[n];

A349445 Dirichlet convolution of A001511 (the 2-adic valuation of 2n) with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2021

sign

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

Cf. A001511, A003602, A349134, A349444 (Dirichlet inverse), A349446 (sum with it).

Cf. also A349432, 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, IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

A001511(n) = (1+valuation(n,2));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349445(n) = sumdiv(n,d,A001511(n/d)*A349134(d));

a(n) = Sum_{d|n} A001511(n/d) * A349134(d).

If p odd prime, a(p) = (1-p)/2. - Bernard Schott, Nov 19 2021

cp's OEIS Frontend

A365393 Lexicographically earliest infinite sequence such that a(i) = a(j) => A364492(i) = A364492(j) for all i, j >= 0, where A364492(n) is the denominator of n / A163511(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A382130 Fractalization of the golden ratio.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A108202 a(2n) = A007376(n); a(2n+1) = a(n).

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A131987 Representation of a dense para-sequence.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A163491 A fractal sequence (if we delete the first occurrence of n we get the sequence itself).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A265104 a(n) = A265100(n+1) - 6, n >= 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A336475 Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (e+1)*p^e.

Views

Author

Keywords

Comments