A003602 -id:A003602 - cp's OEIS frontend

A353368 Dirichlet inverse of A103391, "even fractal sequence".

1, -2, -2, 1, -2, 4, -3, -1, 2, 2, -4, -3, -3, 4, 3, 0, -2, -10, -6, 1, 8, 4, -7, 3, 1, -2, -8, -1, -5, -4, -9, -1, 14, -10, 2, 17, -6, 4, 1, -1, -4, -22, -12, 1, -3, 4, -13, -1, 6, -14, -6, 11, -8, 28, 1, 1, 19, -10, -16, 3, -9, 4, -25, -1, 10, -42, -18, 25, 18, 0, -19, -17, -6, -14, -12, 5, 13, 12, -21, 3, 24

Offset: 1

Antti Karttunen, Apr 18 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003602, A103391, A353369.

Cf. also A349134, A353366.

up_to = 65537;
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) = (n/2^valuation(n, 2)+1)/2; \\ From A003602
A103391(n) = if(1==n,1,(1+A003602(n-1)));
v353368 = DirInverseCorrect(vector(up_to,n,A103391(n)));
A353368(n) = v353368[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A103391(n/d) * a(d).

a(n) = A353369(n) - A103391(n).

A365386 Lexicographically earliest infinite sequence such that a(i) = a(j) => A331410(i) = A331410(j) and A365385(i) = A365385(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 03 2023

Restricted growth sequence transform of the ordered pair [A331410(n), A365385(n)].

For all i, j: A365388(i) = A365388(j) => a(i) = a(j) => A365387(i) = A365387(j).

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

Cf. A003602, A163511, A331410, A365385, A365387, A365388.

Cf. also A334867.

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));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
A365385(n) = A331410(A163511(n));
A365386aux(n) = [A331410(n),A365385(n)];
v365386 = rgs_transform(vector(up_to,n,A365386aux(n)));
A365386(n) = v365386[n];

A365395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365425(i) = A365425(j) and A365427(i) = A365427(j) for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 04 2023

nonn

Restricted growth sequence transform of the ordered pair [A365425(n), A365427(n)].
Restricted growth sequence transform of the function f(n) = A336390(A163511(n)).
For all i, j: a(i) = a(j) => A365385(i) = A365385(j).

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

Cf. A000265, A003602, A046523, A163511, A336390, A365385, A365425, A365427, A365394.

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));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
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));
A365425(n) = A046523(A000265(A163511(n)));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
A365427(n) = A336467(A163511(n));
A365395aux(n) = [A365425(n), A365427(n)];
v365395 = rgs_transform(vector(1+up_to,n,A365395aux(n-1)));
A365395(n) = v365395[1+n];

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

A379006 Ordinal transform of A355582, where A355582 is the largest 5-smooth divisor of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

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

Cf. A355582.
Cf. A379005 (ordinal transform).
Cf. also A003602, A126760.

up_to = 20000;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
is_A051037(n) = (n<7||vecmax(factor(n, 6)[, 1])<7); \\ From A051037
A355582(n) = fordiv(n,d,if(is_A051037(n/d),return(n/d)));
v379006 = ordinal_transform(vector(up_to, n, A355582(n)));
A379006(n) = v379006[n];

A382357 Lexicographically earliest sequence of distinct positive integers such that the 2-adic valuations of adjacent terms differ exactly by one.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 10, 5, 14, 7, 18, 9, 22, 11, 26, 13, 30, 15, 34, 17, 38, 19, 42, 20, 24, 16, 32, 48, 40, 28, 46, 21, 50, 23, 54, 25, 58, 27, 62, 29, 66, 31, 70, 33, 74, 35, 78, 36, 56, 44, 72, 52, 82, 37, 86, 39, 90, 41, 94, 43, 98, 45, 102, 47, 106, 49

Offset: 1

Rémy Sigrist, Mar 22 2025

The first term with a given 2-adic valuation, say k, is necessarily 2^k.
Empirically, powers of two appear as pairs of consecutive terms.
We cannot have three consecutive powers of 2: if a(n) = 2^k and a(n+1) = 2^(k+1) then a(n+2) <= 3*2^k < 2^(k+2).
All powers of two appear in the sequence:
- by contradiction: if 2^m is missing, then the 2-adic valuation of the terms of the sequence is bounded by m,
- by necessity, we have some k < m such that all the integers with 2-adic valuation k appear in the sequence,
- hence all integers with 2-adic valuation k+1 (and k-1 provided k > 0) will appear in the sequence,
- gradually, all integers with 2-adic valuation k+2, k+3, etc. and eventually 2^m, will appear, a contradiction.
Conjecture: this sequence is a permutation of the positive integers.
The fact that A007814 contains every positive integer infinitely many times is not sufficient to guarantee that the present sequence is a permutation of the positive integers (the variant based on A003602 instead of A007814 contains only finitely many even numbers, and so is not a permutation of the positive integers, although A003602 contains every positive integer infinitely many times).

			The initial terms are:
  n   a(n)  A007814(a(n))
  --  ----  -------------
   1     1              0
   2     2              1
   3     3              0
   4     6              1
   5     4              2
   6     8              3
   7    12              2
   8    10              1
   9     5              0
  10    14              1
  11     7              0
  12    18              1
  13     9              0
  14    22              1
  15    11              0

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first million terms (blue pixels for even n's, red pixels for odd n's)
Rémy Sigrist, PARI program

Cf. A003602, A007814, A073675, A266089, A382360.

PARI
```
\\ See Links section.
```

A089265 a(1) = 0; thereafter a(2n) = a(n) + 1, a(2n+1) = 2*n.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 3, 8, 5, 10, 4, 12, 7, 14, 4, 16, 9, 18, 6, 20, 11, 22, 5, 24, 13, 26, 8, 28, 15, 30, 5, 32, 17, 34, 10, 36, 19, 38, 7, 40, 21, 42, 12, 44, 23, 46, 6, 48, 25, 50, 14, 52, 27, 54, 9, 56, 29, 58, 16, 60, 31, 62, 6, 64, 33, 66, 18, 68, 35, 70, 11, 72

Offset: 1

Ralf Stephan, Oct 30 2003

In the binary representation of n, swallow all zeros from the right, then add the number of swallowed zeros, and subtract 1. - Ralf Stephan, Aug 22 2013

First differences of A005766.

Cf. A003602, A220466.

nmax:=73: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := p  + 2*(n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 23 2013

a[n_] := With[{v = IntegerExponent[n, 2]}, v + n/2^v - 1];
Array[a, 100] (* Jean-François Alcover, Feb 28 2019 *)

a(n) = valuation(n,2) + n/2^valuation(n,2) - 1

a(1) = 0; thereafter a(2*n) = a(n) + 1, a(2*n+1) = 2*n.
a(n) = A007814(n) + 2*A025480(n-1) = A007814(n) + A000265(n) - 1.
G.f.: sum(k>=0, (t^2+2t^3-t^4)/(1-t^2)^2, t=(x^2)^k).
a((2*n-1)*2^p) = p + 2*(n-1), p >= 0. - Johannes W. Meijer, Jan 23 2013

A110962 Fractalization of A025480, zero-based version of Kimberling's paraphrases sequence.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 3, 0, 0, 0, 4, 2, 2, 1, 5, 1, 1, 0, 6, 3, 3, 0, 7, 0, 0, 0, 8, 4, 4, 2, 9, 2, 2, 1, 10, 5, 5, 1, 11, 1, 1, 0, 12, 6, 6, 3, 13, 3, 3, 0, 14, 7, 7, 0, 15, 0, 0, 0, 16, 8, 8, 4, 17, 4, 4, 2, 18, 9, 9, 2, 19, 2, 2, 1, 20, 10, 10, 5, 21, 5, 5, 1, 22, 11, 11, 1, 23, 1, 1, 0, 24, 12, 12

Offset: 0

Alexandre Wajnberg, Sep 26 2005

Self-descriptive sequence: the terms at odd indices are the sequence itself, while the terms at even indices (the skeleton of this sequence) are the terms of A025480, which is a zero-based sequence of Kimberling's paraphrases sequence, A003602.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Clark Kimberling, Fractal sequences
Index entries for sequences related to binary expansion of n

Cf. A000265, A003602, A025480, A089309, A103391, A110812, A110779, A110766.

One less than A110963 (note also the different starting offsets).

A025480(n) = (n>>valuation(n*2+2, 2));
A110962(n) = if(!(n%2), A025480(n/2), A110962((n-1)/2)); \\ Antti Karttunen, Apr 18 2022

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

For even n, a(n) = A025480(n/2), for odd n, a(n) = a((n-1)/2). - Antti Karttunen, Apr 18 2022
a(2n+1) = a(4n+3) = a(n).
a(2n) = a(4n+1) = a(4n+2) = A025480(n/2).
a(4n) = a(8n+1) = a(8n+2) = n.
a(n) = A110963(1+n) - 1.

Entry edited and more terms added by Antti Karttunen, Apr 18 2022

A123390 Triangle read by rows: n-th row starts with n and continues with half the previous value as long as that is even.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Oct 13 2006

A fractal sequence, generated by the rule a(n) is a new maximum when a(n-1) is odd and a repetition of an earlier value when a(n-1) is even.
From Flávio V. Fernandes, Mar 13 2025: (Start)
a(n) is given by A003602(n) at A001511(n) diagram
  1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9
  . 1 . . . 2 . . . 3 . . . 4 . . .
  . . . 1 . . . . . . . 2 . . . . .
  . . . . . . . 1 . . . . . . . . .
  . . . . . . . . . . . . . . . 1 .
read by backwards 2^n, which is given by A118319(n) at A001511(n) diagram
  1 . 2 . 4 . 5 . 8 . 9 .11 .12 .16
  . 3 . . . 6 . . .10 . . .13 . . .
  . . . 7 . . . . . . .14 . . . . .
  . . . . . . .15 . . . . . . . . .
  . . . . . . . . . . . . . . .31 . - see formula. (End)

			Triangle starts
  1;
  2, 1;
  3;
  4, 2, 1;
  5;
  6, 3;
  7;
  8, 4, 2, 1;
  9;
  10, 5;
  11;
  12, 6, 3;
  13;

Alois P. Heinz, Rows n = 1..10000, flattened

Row lengths are A001511.
Row sums give A129527.
Cf. A120385.
Cf. A003602, A108918, A118319.

T:= proc(n) local m,l; m:=n; l:= m;
      while irem(m, 2, 'm')=0 do l:=l,m od: l
    end:
seq(T(n), n=1..40);  # Alois P. Heinz, Oct 09 2015

Flatten[Function[n, NestWhile[Append[#, Last[#]/2] &, {n}, EvenQ[Last[#]] &]][#] & /@ Range[20]] (* Birkas Gyorgy, Apr 13 2011 *)

a(1) = 1, for n > 1, if a(n-1) is even, a(n) = a(n-1)/2, otherwise a(n) = (max_{k

Ordinal transform of A082850.

a(n) = A003602(A108918(n)). - Flávio V. Fernandes, Mar 13 2025

A336392 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A336467(i) = A336467(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, 3, 9, 6, 5, 17, 10, 18, 3, 19, 11, 20, 6, 21, 12, 22, 2, 23, 13, 24, 7, 25, 14, 26, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 3, 32, 9, 33, 6, 34, 5, 35, 17, 36, 10, 21, 18, 37, 3, 11, 19, 38, 11, 39, 20, 40, 6, 41, 21, 42, 12, 43, 22, 44, 2, 45, 23, 46, 13

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the ordered pair [A278222(n), A336467(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. A003602, A278222, A324400, A336390, A336391, A336394, A336467.

Cf. also A286622, A336472.

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; };
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));
A000265(n) = (n>>valuation(n,2));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
Aux336392(n) = [A278222(n), A336467(n)];
v336392 = rgs_transform(vector(up_to, n, Aux336392(n)));
A336392(n) = v336392[n];

A336393 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336467(i) = A336467(j) and A278221(A000265(i)) = A278221(A000265(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the ordered pair [A336467(n), A278221(A000265(n))], or equally, of the ordered pair [A336467(n), A336395(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, A324400, A278221, A286621, A336390, A336395, A336467.

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));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
Aux336393(n) = [A336467(n), A278221(A000265(n))];
v336393 = rgs_transform(vector(up_to, n, Aux336393(n)));
A336393(n) = v336393[n];

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A353368 Dirichlet inverse of A103391, "even fractal sequence".

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A365386 Lexicographically earliest infinite sequence such that a(i) = a(j) => A331410(i) = A331410(j) and A365385(i) = A365385(j) for all i, j >= 1.

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A365395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365425(i) = A365425(j) and A365427(i) = A365427(j) for all i, j >= 0.

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A379006 Ordinal transform of A355582, where A355582 is the largest 5-smooth divisor of n.

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A382357 Lexicographically earliest sequence of distinct positive integers such that the 2-adic valuations of adjacent terms differ exactly by one.

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A089265 a(1) = 0; thereafter a(2*n) = a(n) + 1, a(2*n+1) = 2*n.

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Maple

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A110962 Fractalization of A025480, zero-based version of Kimberling's paraphrases sequence.

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A123390 Triangle read by rows: n-th row starts with n and continues with half the previous value as long as that is even.

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A089265 a(1) = 0; thereafter a(2n) = a(n) + 1, a(2n+1) = 2*n.