A003602 -id:A003602 - cp's OEIS frontend

A336390 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336467(i) = A336467(j) and A336158(i) = A336158(j), for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Aug 10 2020

nonn

Restricted growth sequence transform of the ordered pair [A336467(n), A336158(n)].
For all i, j:
  A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j),
  a(i) = a(j) => A331410(i) = A331410(j),
  a(i) = a(j) => A336391(i) = A336391(j).

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

Cf. A003602, A324400, A336158, A336467, A336391, A336393.

Cf. also A336470.

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
A336158(n) = A046523(A000265(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])); };
Aux336390(n) = [A336158(n), A336467(n)];
v336390 = rgs_transform(vector(up_to, n, Aux336390(n)));
A336390(n) = v336390[n];

A110766 Fractalization of Pi.

Original entry on oeis.org

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

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 Pi.

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

Cf. A000796 (Pi), A003602.

Cf. A110779 (of e), A110812 (of sqrt 2), A382130 (of phi).

import Data.List (transpose)
a110766 n = a110766_list !! (n-1)
a110766_list = concat $ transpose [a000796_list, a110766_list]
-- Reinhard Zumkeller, Aug 29 2014

a(2n) = a(n); a(2n-1) = digits of Pi.

a(85) corrected and formula fixed by Reinhard Zumkeller, Aug 29 2014

A117303 Self-inverse permutation of the natural numbers based on the bijection (2x-1)2^(y-1) <--> (2y-1)2^(x-1).

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 8, 7, 16, 12, 32, 10, 64, 24, 128, 9, 256, 48, 512, 20, 1024, 96, 2048, 14, 4096, 192, 8192, 40, 16384, 384, 32768, 11, 65536, 768, 131072, 80, 262144, 1536, 524288, 28, 1048576, 3072, 2097152, 160, 4194304, 6144, 8388608, 18, 16777216

Offset: 1

Reinhard Zumkeller, Apr 24 2006

nonn

a(a(n)) = n; fixed points A014480: a(A014480(n)) = A014480(n). - Reinhard Zumkeller, Apr 27 2006

Cf. A000265, A007814, A006519, A005408, A000079.

Cf. A118413, A118416.

a:= n-> (j-> (2*j+1)*2^((n/2^j-1)/2))(padic[ordp](n, 2)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 23 2019

a[n_] := (2 IntegerExponent[2 n, 2] - 1)*2^((n/2^IntegerExponent[n, 2] + 1)/2 - 1); Array[a, 50] (* Jean-François Alcover, Mar 12 2019 *)

def A117303(n): return (((m:=(n&-n).bit_length())<<1)-1)*(1<<(n>>m)) # Chai Wah Wu, Jul 14 2022

a(n) = (2*A001511(n) - 1) * 2^(A003602(n) - 1).

Spelling corrected by Jason G. Wurtzel, Aug 23 2010

A330896 Lexicographically earliest sequence of positive integers such that for any m > 0, gaps between consecutive m's are all distinct.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, May 01 2020

Every positive integer appears infinitely many times in the sequence.
This sequence has similarities with A003602, where gaps between consecutive equal values are all distinct.
This sequence has similarities with A002260, where for any m > 0, gaps between consecutive m's are strictly increasing.
Apparently, for any m > 0:
- the k-th gap between consecutive m's equals k except for finitely many k's,
- the k-th occurrence of m appears at index A330897(m) + A000217(k-1) except for finitely many k's.

			The first terms, alongside the gaps for m = 1..4, are:
     n|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  ...
  a(n)|  1  1  2  1  2  2  1  3  2  3  1  4  2  3  3  1  4  2  4  3  5  1  3  ...
  ----+---------------------------------------------------------------------
   1's|     1,    2,       3,          4,             5,                6,    ...
   2's|              2, 1,       3,          4,             5,                ...
   3's|                             2,          4, 1,             5,       3, ...
   4's|                                                  5,    2,             ...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A330896

Cf. A000124, A000217, A002260, A003602, A330897.

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

from itertools import islice
def agen(): # generator of terms
    m, a = dict(), []
    while True:
        an, allnew = 0, False
        while not allnew:
            allnew, an, mn = True, an+1, set()
            for i in range(len(a)-1, -1, -1):
                if an == a[i]:
                    t = len(a)-i+1
                    if (an in m and t in m[an]) or t in mn: allnew = False; break
                    mn.add(t)
                    break
        yield an; a.append(an)
        if an not in m: m[an] = set()
        m[an] |= mn
print(list(islice(agen(), 87))) # Michael S. Branicky, Dec 06 2024

a(n) = 1 iff n belongs to A000124.

A336159 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

Restricted growth sequence transform of the ordered pair [A278222(n), A336158(n)], i.e., of the ordered pair [A046523(A005940(1+n)), A046523(A000265(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, A005940, A046523, A278222, A336158.

Cf. also A003602, A286163, A286622, A291762, A324400, A336149, A336156, A336157, A336160, A336162.

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));
Aux336159(n) = [A278222(n), A336158(n)];
v336159 = rgs_transform(vector(up_to, n, Aux336159(n)));
A336159(n) = v336159[n];

A337226 Lexicographically earliest sequence of positive integers with the property that, for all k > 0, there is at most one j such that a(j) = a(j+k).

Original entry on oeis.org

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

Offset: 1

Samuel B. Reid, Aug 19 2020

The sequence initially appears to be trivially fractal in that the removal of the first occurrence of each value seems to yield the original sequence. This pattern continues until a(121) where, if the sequence were fractal in this way, the value would be 72 or 1. The actual value is 13, so the pattern is broken.

Conjecture: For all k > 0, there is exactly one j such that a(j) = a(j+k). For 0 < k < 11911, this conjecture holds.

			  1 1 2 1 3 4 2
   (1)1 2 1 3 4   k = 1
      1(1)2 1 3   k = 2
       (1)1 2 1   k = 3
          1 1(2)  k = 4
            1 1   k = 5
              1   k = 6
Coincidences are circled. There can only be one coincidence per row.
a(3) cannot be 1 because that would result in two coincidences for k = 1.
a(5) cannot be 1 or 2 because those values would result in two coincidences for k = 1 and k = 2, respectively.
a(7) cannot be 1, 3, or 4 because those values would result in two coincidences for k = 3, k = 2, and k = 1, respectively. It can, however, be 2 because this results in no double coincidences.

Samuel B. Reid, Table of n, a(n) for n = 1..10000
Samuel B. Reid, Python program for A337226

Cf. A003602, A014552, A026272.

Python
```
# See Links section.
```

A249725 Inverse permutation to A135764.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 7, 10, 11, 8, 16, 9, 22, 12, 29, 15, 37, 17, 46, 13, 56, 23, 67, 14, 79, 30, 92, 18, 106, 38, 121, 21, 137, 47, 154, 24, 172, 57, 191, 19, 211, 68, 232, 31, 254, 80, 277, 20, 301, 93, 326, 39, 352, 107, 379, 25, 407, 122, 436, 48, 466, 138, 497, 28, 529, 155, 562, 58, 596, 173, 631, 32, 667, 192, 704, 69, 742, 212, 781, 26, 821, 233, 862, 81

Offset: 1

Antti Karttunen, Nov 15 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A135764.
Similar or related permutations: A209268, A246276, A246676, A249742, A249811.
Cf. A000079, A000124, A000217, A003602, A005408, A007814.

a(n) = {r = 1; while(!(n%2), n = n >> 1; r++); k = 1 + n \ 2; binomial(r + k - 1, 2) + r} \\ David A. Corneth, Feb 05 2015

(define (A249725 n) (+ 1 (* (/ 1 2) (+ (expt (+ (A003602 n) (A007814 n)) 2) (- (A003602 n)) (A007814 n)))))

a(n) = 1 + (((A003602(n)+A007814(n))^2 + A007814(n) - A003602(n))/2).
As a composition of other permutations:
a(n) = A249742(A249811(n)).
a(n) = A246276(A246676(n)).
Other identities. For all n >= 0 the following holds:
a(A005408(n)) = A000124(n). [Maps odd numbers to central polygonal numbers].
a(A000079(n)) = A000217(n+1). [Maps powers of two to triangular numbers].

A336471 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 22 2020

nonn

Restricted growth sequence transform of the ordered pair [A329697(n), A336158(n)].
For all i, j:
  A336470(i) = A336470(j) => a(i) = a(j)
  a(i) = a(j) => A336396(i) = A336396(j),
  a(i) = a(j) => A336469(i) = A336469(j) => A336477(i) = A336477(j).
This sequence has an ability to see where the terms of A003401 are, as they are the indices of zeros in A336469. Specifically, they are numbers k that satisfy the condition A329697(k) = A001221(A336158(k)), i.e., numbers for which A329697(k) is equal to the number of distinct prime divisors of the odd part of k. See also comments in array A334100.

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

Cf. A000265, A003401, A046523, A329697, A334100, A336158.

Cf. also A003602, A324400, A336159, A336160, A336396, A336469, A336470, A336472, A336473, A336477.

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
A336158(n) = A046523(A000265(n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
Aux336471(n) = [A329697(n), A336158(n)];
v336471 = rgs_transform(vector(up_to, n, Aux336471(n)));
A336471(n) = v336471[n];

A365431 Lexicographically earliest infinite sequence such that a(i) = a(j) => A364502(i) = A364502(j) for all i, j >= 1, where A364502(n) is the denominator of n / A005940(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 07 2023

Restricted growth sequence transform of A364502, or equally, of A365432.
For all i, j: A003602(i) = A003602(j) => a(i) = a(j).
Compare to the scatter plots of A365393 and A365715.

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

Cf. A005940, A364502, A365432.

Cf. also A365393, A365715 (analogous sequence for Doudna variant D(3)).

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 };
A364502(n) = { my(u=A005940(n)); (u / gcd(n, u)); };
v365431 = rgs_transform(vector(up_to,n,A364502(n)));
A365431(n) = v365431[n];

A110779 Fractalization of e.

Original entry on oeis.org

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

Clark Kimberling, Fractal sequences.

Cf. A001113, A003602.

Cf. A110766 (of Pi), A110812 (of sqrt 2), A382130 (of phi).

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

cp's OEIS Frontend

A336390 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336467(i) = A336467(j) and A336158(i) = A336158(j), for all i, j >= 1.

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A110766 Fractalization of Pi.

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A117303 Self-inverse permutation of the natural numbers based on the bijection (2*x-1)*2^(y-1) <--> (2*y-1)*2^(x-1).

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A330896 Lexicographically earliest sequence of positive integers such that for any m > 0, gaps between consecutive m's are all distinct.

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A336159 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A336158(i) = A336158(j), for all i, j >= 1.

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A337226 Lexicographically earliest sequence of positive integers with the property that, for all k > 0, there is at most one j such that a(j) = a(j+k).

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A249725 Inverse permutation to A135764.

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A336471 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A336158(i) = A336158(j), for all i, j >= 1.

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A117303 Self-inverse permutation of the natural numbers based on the bijection (2x-1)2^(y-1) <--> (2y-1)2^(x-1).