A332817 -id:A332817 - cp's OEIS frontend

A163511 a(0)=1. a(n) = p(A000120(n)) * Product_{m=1..A000120(n)} p(m)^A163510(n,m), where p(m) is the m-th prime.

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 25, 12, 15, 10, 7, 32, 81, 54, 125, 36, 75, 50, 49, 24, 45, 30, 35, 20, 21, 14, 11, 64, 243, 162, 625, 108, 375, 250, 343, 72, 225, 150, 245, 100, 147, 98, 121, 48, 135, 90, 175, 60, 105, 70, 77, 40, 63, 42, 55, 28, 33, 22, 13, 128

Offset: 0

Leroy Quet, Jul 29 2009

This is a permutation of the positive integers.
From Antti Karttunen, Jun 20 2014: (Start)
Note the indexing: the domain starts from 0, while the range excludes zero, thus this is neither a bijection on the set of nonnegative integers nor on the set of positive natural numbers, but a bijection from the former set to the latter.
Apart from that discrepancy, this could be viewed as yet another "entanglement permutation" where the two complementary pairs to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with the complementary pair even numbers (taken straight) and odd numbers in the order they appear in A003961: (A005843/A003961). See also A246375 which has almost the same recurrence.
Note how the even bisection halved gives the same sequence back. (For a(0)=1, take ceiling of 1/2).
(End)
From Antti Karttunen, Dec 30 2017: (Start)
This irregular table can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A003961 to the parent:
                                     1
                                     |
                  ...................2...................
                 4                                       3
       8......../ \........9                   6......../ \........5
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  16       27         18       25         12       15         10       7
32  81   54  125    36  75   50  49     24  45   30  35     20  21   14 11
etc.
Sequence A005940 is obtained by scanning the same tree level by level in mirror image fashion. Also in binary trees A253563 and A253565 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) gives the distance of n from 1 in all these trees, and A252463 gives the parent of the node containing n.
A252737(n) gives the sum and A252738(n) the product of terms on row n (where 1 is on row 0, 1 on row 1, 3 and 4 on row 2, etc.). A252745(n) gives the number of nodes on level n whose left child is smaller than the right child, and A252744(n) is an indicator function for those nodes.
(End)
Note that the idea behind maps like this (and the mirror image A005940) admits also using alternative orderings of primes, not just standard magnitude-wise ordering (A000040). For example, A332214 is a similar sequence but with primes rearranged as in A332211, and A332817 is obtained when primes are rearranged as in A108546. - Antti Karttunen, Mar 11 2020
From Lorenzo Sauras Altuzarra, Nov 28 2020: (Start)
This sequence is generated from A228351 by applying the following procedure: 1) eliminate the compositions that end in one unless the first one, 2) subtract one unit from every component, 3) replace every tuple [t_1, ..., t_r] by Product_{k=1..r} A000040(k)^(t_k) (see the examples).
Is it true that a(n) = A337909(n+1) if and only if a(n+1) is not a term of A161992?
Does this permutation have any other cycle apart from (1), (2) and (6, 9, 16, 7)? (End)
From Antti Karttunen, Jul 25 2023: (Start)
(In the above question, it is assumed that the starting offset would be 1 instead of 0).
Questions:
Does a(n) = 1+A054429(n) hold only when n is of the form 2^k times 1, 3 or 7, i.e., one of the terms of A029748?
It seems that A007283 gives all fixed points of map n -> a(n), like A335431 seems to give all fixed points of map n -> A332214(n). Is there a general rule for mappings like these that the fixed points (if they exist) must be of the form 2^k times a certain kind of prime, i.e., that any odd composite (times 2^k) can certainly be excluded? See also note in A029747.
(End)
If the conjecture given in A364297 holds, then it implies the above conjecture about A007283. See also A364963. - Antti Karttunen, Sep 06 2023
Conjecture: a(n^k) is never of the form x^k, for any integers n > 0, k > 1, x >= 1. This holds at least for squares, cubes, seventh and eleventh powers (see A365808, A365801, A366287 and A366391). - Antti Karttunen, Sep 24 2023, Oct 10 2023.
See A365805 for why the above holds for any n^k, with k > 1. - Antti Karttunen, Nov 23 2023

			For n=3, whose binary representation is "11", we have A000120(3)=2, with A163510(3,1) = A163510(3,2) = 0, thus a(3) = p(2) * p(1)^0 * p(2)^0 = 3*1*1 = 3.
For n=9, "1001" in binary, we have A000120(9)=2, with A163510(9,1) = 0 and A163510(9,2) = 2, thus a(9) = p(2) * p(1)^0 * p(2)^2 = 3*1*9 = 27.
For n=10, "1010" in binary, we have A000120(10)=2, with A163510(10,1) = 1 and A163510(10,2) = 1, thus a(10) = p(2) * p(1)^1 * p(2)^1 = 3*2*3 = 18.
For n=15, "1111" in binary, we have A000120(15)=4, with A163510(15,1) = A163510(15,2) = A163510(15,3) = A163510(15,4) = 0, thus a(15) = p(4) * p(1)^0 * p(2)^0 * p(3)^0 * p(4)^0 = 7*1*1*1*1 = 7.
[1], [2], [1,1], [3], [1,2], [2,1] ... -> [1], [2], [3], [1,2], ... -> [0], [1], [2], [0,1], ... -> 2^0, 2^1, 2^2, 2^0*3^1, ... = 1, 2, 4, 3, ... - _Lorenzo Sauras Altuzarra_, Nov 28 2020

Inverse: A243071.
Cf. A000040, A000120, A000225, A000788, A003961, A007814, A054429, A055396, A064216, A135523, A161992, A163510, A245605, A245612, A246375, A246378, A246681, A161511, A228351, A243499, A243503, A243504, A269854, A280873, A285727, A290251, A293437, A337909.
Cf. A007283 (known positions where a(n)=n), A029747, A029748, A364255 [= gcd(n,a(n))], A364258 [= a(n)-n], A364287 (where a(n) < n), A364292 (where a(n) <= n), A364494 (where n|a(n)), A364496 (where a(n)|n), A364963, A364297.
Cf. A365808 (positions of squares), A365801 (of cubes), A365802 (of fifth powers), A365805 [= A052409(a(n))], A366287, A366391.
Cf. A005940, A332214, A332817, A366275 (variants).

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~
Table[Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}]][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2016 *)

from sympy import prime
def A163511(n):
    if n:
        k, c, m = n, 0, 1
        while k:
            c += 1
            m *= prime(c)**(s:=(~k&k-1).bit_length())
            k >>= s+1
        return m*prime(c)
    return 1 # Chai Wah Wu, Jul 17 2023

For n >= 1, a(2n) is even, a(2n+1) is odd. a(2^k) = 2^(k+1), for all k >= 0.
From Antti Karttunen, Jun 20 2014: (Start)
a(0) = 1, a(1) = 2, a(2n) = 2*a(n), a(2n+1) = A003961(a(n)).
As a more general observation about the parity, we have:
For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [This permutation preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]
For n >= 1, A055396(a(n)) = A091090(n) = A007814(n+1) + 1 - A036987(n).
For n >= 1, a(A000225(n)) = A000040(n).
(End)
From Antti Karttunen, Oct 11 2014: (Start)
As a composition of related permutations:
a(n) = A005940(1+A054429(n)).
a(n) = A064216(A245612(n))
a(n) = A246681(A246378(n)).
Also, for all n >= 0, it holds that:
A161511(n) = A243503(a(n)).
A243499(n) = A243504(a(n)).
(End)
More linking identities from Antti Karttunen, Dec 30 2017: (Start)
A046523(a(n)) = A278531(n). [See also A286531.]
A278224(a(n)) = A285713(n). [Another filter-sequence.]
A048675(a(n)) = A135529(n) seems to hold for n >= 1.
A250245(a(n)) = A252755(n).
A252742(a(n)) = A252744(n).
A245611(a(n)) = A253891(n).
A249824(a(n)) = A275716(n).
A292263(a(n)) = A292264(n). [A292944(n) + A292264(n) = n.]
--
A292383(a(n)) = A292274(n).
A292385(a(n)) = A292271(n). [A292271(n) +  A292274(n) = n.]
--
A292941(a(n)) = A292942(n).
A292943(a(n)) = A292944(n).
A292945(a(n)) = A292946(n). [A292942(n) + A292944(n) + A292946(n) = n.]
--
A292253(a(n)) = A292254(n).
A292255(a(n)) = A292256(n). [A292944(n) + A292254(n) + A292256(n) = n.]
--
A279339(a(n)) = A279342(n).
a(A071574(n)) = A269847(n).
a(A279341(n)) = A279338(n).
a(A252756(n)) = A250246(n).
(1+A008836(a(n)))/2 = A059448(n).
(End)
From Antti Karttunen, Jul 26 2023: (Start)
For all n >= 0, a(A007283(n)) = A007283(n).
A001222(a(n)) = A290251(n).
(End)

More terms computed and examples added by Antti Karttunen, Jun 20 2014

A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 73, 68, 87, 70, 67, 72

Offset: 1

Reinhard Zumkeller, Jun 10 2005

Multiplicative with a(2^e) = 2^e, else if p is the m-th prime then a(p^e) = q^e where q is the m/2-th prime of the form 4*k + 3 (A002145) for even m and a(p^e) = r^e where r is the (m-1)/2-th prime of the form 4*k + 1 (A002144) for odd m. - David A. Corneth, Apr 25 2022

Permutation of the natural numbers with fixed points A108549: a(A108549(n)) = A108549(n).

Antti Karttunen, Table of n, a(n) for n = 1..26927
Index entries for sequences that are permutations of the natural numbers

Cf. A002144, A002145, A049084, A108546, A108549 (fixed points), A332808 (inverse permutation).
Cf. also A332815, A332817 (this permutation applied to Doudna tree and its mirror image), also A332818, A332819.
Cf. also A267099, A332212 and A348746 for other similar mappings.

terms = 72;
A111745 = Module[{prs = Prime[Range[2 terms]], m3, m1, min},
     m3 = Select[prs, Mod[#, 4] == 3&];
     m1 = Select[prs, Mod[#, 4] == 1&];
     min = Min[Length[m1], Length[m3]];
     Riffle[Take[m3, min], Take[m1, min]]];
A108546[n_] := If[n == 1, 2, A111745[[n - 1]]];
A049084[n_] := PrimePi[n]*Boole[PrimeQ[n]];
a[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe; A108546[A049084[p]]^e, {pe, FactorInteger[n]}]]];
Array[a, terms] (* Jean-François Alcover, Nov 19 2021, using Harvey P. Dale's code for A111745 *)

up_to = 26927; \\ One of the prime fixed points.
A108546list(up_to) = { my(v=vector(up_to), p,q); v[1] = 2; v[2] = 3; v[3] = 5; for(n=4,up_to, p = v[n-2]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[n] = q); (v); };
v108546 = A108546list(up_to);
A108546(n) = v108546[n];
A108548(n) = { my(f=factor(n)); f[,1] = apply(A108546,apply(primepi,f[,1])); factorback(f); }; \\ Antti Karttunen, Apr 25 2022

Name edited by Antti Karttunen, Apr 25 2022

A365805 a(n) = largest exponent m for which a representation of the form A163511(n) = k^m exists (for some k). a(0) = 0 by convention.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 01 2023

nonn

Equivalently, the largest exponent m for which a representation of the form A332214(n) = k^m exists (for some k), or similarly, for any other such variant of A163511, like A332817.

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

Cf. A052409, A163511, A332214, A332817, A365807, A366370.

Cf. A365808 (positions of even terms), A365801 (multiples of 3), A365802 (multiples of 5), A366287 (multiples of 7), A366391 (multiples of 11).

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); };
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));
A365805(n) = A052409(A163511(n));

a(n) = A052409(A163511(n)).

If a(n) > 1 (or A052409(n) > 1), then a(n) <> A052409(n). [Consider A366370]

A365802 Numbers k such that A163511(k) is a fifth power.

Original entry on oeis.org

0, 16, 33, 67, 135, 271, 512, 543, 1025, 1056, 1087, 2051, 2113, 2144, 2175, 4103, 4227, 4289, 4320, 4351, 8207, 8455, 8579, 8641, 8672, 8703, 16384, 16415, 16911, 17159, 17283, 17345, 17376, 17407, 32769, 32800, 32831, 33792, 33823, 34319, 34567, 34691, 34753, 34784, 34815, 65539, 65601, 65632, 65663, 67585, 67616

Offset: 1

Antti Karttunen, Oct 01 2023

nonn

Equivalently, numbers k for which A332214(k), and also A332817(k) are fifth powers.
The sequence is defined inductively as:
 (a) it contains 0 and 16, and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 32*a(n) are also included as terms.
When iterating n -> 2n+1 mod 31, starting from 16 we obtain five distinct remainders 16, 2, 5, 11, 23, before the cycle starts again from 16. (see A153893), while x^5 mod 31 may obtain only these values: 0, 1, 5, 6, 25, 26, 30. The only common element of these sets is 5. We have x^5 == 5 (mod 31) whenever x == 7, 14, 19, 25, 28 mod 31, with all other x leaving a remainder that is not in the set [16, 2, 5, 11, 23].
On the other hand, when iterating n -> 2n+1 mod 33, starting from 16 we obtain ten distinct remainders 16, 0, 1, 3, 7, 15, 31, 30, 28, 24, before the cycle starts again from 16, while x^5 mod 33 obtain only these values: 0, 1, 10, 11, 12, 21, 22, 23, 32. We have x^5 == 0 (mod 33) iff x == 0 (mod 33) and x^5 == 1 (mod 33) whenever x == 1, 4, 16, 25, 31 mod 33. In the n->2n+1 cycles of 5 and 10 elements starting from 16, the 5's (of every second cycle) in the former and the 1's in the latter are aligned with each other.
In any case, this sequence do not contain any fifth powers after the initial zero. See A365805. - Antti Karttunen, Nov 23 2023

Index entries for sequences related to binary expansion of n

Positions of multiples of 5 in A365805.
Sequence A243071(n^5), n >= 1, sorted into ascending order.
Subsequences: A013825, A198275.
Cf. A000584, A153893, A163511, A332214, A332817.
Cf. also A365801, A365808, A366287, A366391.

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));
isA365802(n) = ispower(A163511(n),5);

isA365802(n) = if(n<=16, !(n%16), if(n%2, isA365802((n-1)/2), if(n%32, 0, isA365802(n/32))));

A332811 a(n) = A243071(A332808(n)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 63, 12, 31, 30, 13, 8, 127, 10, 255, 28, 29, 126, 1023, 24, 11, 62, 9, 60, 511, 26, 4095, 16, 125, 254, 27, 20, 2047, 510, 61, 56, 8191, 58, 16383, 252, 25, 2046, 65535, 48, 23, 22, 253, 124, 32767, 18, 123, 120, 509, 1022, 262143, 52, 131071, 8190, 57, 32, 59, 250, 1048575, 508, 2045, 54, 4194303, 40, 524287, 4094, 21

Offset: 1

Antti Karttunen, Mar 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Cf. A332817 (inverse permutation).
Cf. A054429, A243071, A332808, A332816, A332895, A332896, A332899.
Cf. also A332215.

up_to = 26927;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };
A332811(n) = A243071(A332808(n));

a(n) = A243071(A332808(n)).
For n > 1, a(n) = A054429(A332816(n)).
a(n) = A332895(n) + A332896(n).
a(n) = A332895(n) OR A332896(n) = A332895(n) XOR A332896(n).
A000120(a(n)) = A332899(n).

A366287 Numbers k such that A163511(k) is a seventh power.

Original entry on oeis.org

0, 64, 129, 259, 519, 1039, 2079, 4159, 8192, 8319, 16385, 16512, 16639, 32771, 33025, 33152, 33279, 65543, 66051, 66305, 66432, 66559, 131087, 132103, 132611, 132865, 132992, 133119, 262175, 264207, 265223, 265731, 265985, 266112, 266239, 524351, 528415, 530447, 531463, 531971, 532225, 532352, 532479, 1048576, 1048703

Offset: 1

Antti Karttunen, Oct 09 2023

nonn

Equivalently, numbers k for which A332214(k), and also A332817(k) are seventh powers.
The sequence is defined inductively as:
 (a) it contains 0 and 64,
   and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 128*a(n) are also included as terms.
When iterating n -> 2n+1 mod 127, starting from 64 we get 64, 2, 5, 11, 23, 47, 95, and then cycle starts again from 64 (see A153893), while on the other hand, x^7 mod 127 obtains values: 0, 1, 19, 20, 22, 24, 28, 37, 52, 59, 68, 75, 90, 99, 103, 105, 107, 108, 126. These sets have no terms in common, therefore there are no seventh powers in this sequence after the initial 0.

Positions of multiples of 7 in A365805.
Sequence A243071(n^7), n >= 1, sorted into ascending order.
Cf. A001015, A153893, A163511, A332214, A332817.
Cf. A365801, A365802, A365808, A366391.

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));
isA366287(n) = ispower(A163511(n),7);

isA366287(n) = if(n<=64, !(n%64), if(n%2, isA366287((n-1)/2), if(n%128, 0, isA366287(n>>7))));

A366391 Numbers k such that A163511(k) is an eleventh power.

Original entry on oeis.org

0, 1024, 2049, 4099, 8199, 16399, 32799, 65599, 131199, 262399, 524799, 1049599, 2097152, 2099199, 4194305, 4196352, 4198399, 8388611, 8392705, 8394752, 8396799, 16777223, 16785411, 16789505, 16791552, 16793599, 33554447, 33570823, 33579011, 33583105, 33585152, 33587199, 67108895, 67141647, 67158023, 67166211, 67170305

Offset: 1

Antti Karttunen, Oct 09 2023

nonn

Equivalently, numbers k for which A332214(k), and also A332817(k) are eleventh powers.
The sequence is defined inductively as:
 (a) it contains 0 and 1024,
   and
 (b) for any nonzero term a(n), (2*a(n)) + 1 and 2048*a(n) are also included as terms.
When iterating n -> 2n+1 mod 2047, starting from 1024 we get 1024, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535,  and then cycle starts again from 1024 (see A153893), while on the other hand, x^11 mod 2047 obtains values: 0, 1, 230, 322, 344, 368, 390, 482, 622, 712, 713, 942, 967, 1013, 1034, 1080, 1105, 1334, 1335, 1425, 1565, 1657, 1679, 1703, 1725, 1817, 2046. These sets have no terms in common, therefore there are no eleventh powers in this sequence after the initial 0.

Positions of multiples of 11 in A365805.
Sequence A243071(n^11), n >= 1, sorted into ascending order.
Cf. A008455, A153893, A163511, A332214, A332817.
Cf. also A365801, A365802, A365808, A366287.

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));
isA366391v(n) = ispower(A163511(n),11);

isA366391(n) = if(n<=1024, !(n%1024), if(n%2, isA366391((n-1)/2), if(n%2048, 0, isA366391(n>>11))));

cp's OEIS Frontend

A332995 a(n) = A332895(A332817(n)).

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A332996 a(n) = A332896(A332817(n)).

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A163511 a(0)=1. a(n) = p(A000120(n)) * Product_{m=1..A000120(n)} p(m)^A163510(n,m), where p(m) is the m-th prime.

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A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4*k+1 and 4*k+3 alternate, and prime(j) is the j-th prime in A000040.

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A365805 a(n) = largest exponent m for which a representation of the form A163511(n) = k^m exists (for some k). a(0) = 0 by convention.

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A365802 Numbers k such that A163511(k) is a fifth power.

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A332811 a(n) = A243071(A332808(n)).

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A366287 Numbers k such that A163511(k) is a seventh power.

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A108548 Fully multiplicative with a(prime(j)) = A108546(j), where A108546 is the lexicographically earliest permutation of primes such that after 2 the forms 4k+1 and 4k+3 alternate, and prime(j) is the j-th prime in A000040.