A289814 -id:A289814 - cp's OEIS frontend

A296073 Filter combining A296071(n) and A296072(n), related to the deficiencies of proper divisors of n.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 33

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Construction: Pack the values of A296071(n) and A296072(n) to a single value with any injective N x N -> N packing function, like for example as f(n) = (1/2)*(2 + ((A296071(n)+A296072(n))^2) - A296071(n) - 3*A296072(n)) (the packing function here is the two-argument form of A000027). Then apply the restricted growth sequence transform to the sequence f(1), f(2), f(3), ... The transform assigns a unique increasing number for each newly encountered term of the sequence, and for any subsequent occurrences of the same term it gives the same number that term obtained for the first time.
For all i, j: a(i) = a(j) => A296074(i) = A296074(j).
Note that this is NOT restricted growth transform of A239968, which is A305800. Apart from 2's that occur at every prime, there are other duplicates also, first at a(125) = a(46) = 33.

			To see that a(46) and a(125) have the same value (33), consider the proper divisors of 46 = 1, 2, 23 and of 125 = 1, 5, 25. Their deficiencies are 1, 1, 22 and 1, 4, 19 respectively. When we look at their balanced ternary representations [as here all elements are positive, it can be obtained as A007089(A117967(n)) with 2's standing for -1's]:
   1 =    1
   1 =    1
  22 = 1211 (as 22 = 1*(3^3) + -1*(3^2) + 1*(3^1) + 1*(3^0))
and
   1 =    1
   4 =   11
  19 = 1201 (as 19 = 1*(3^3) + -1*(3^2) + 0*(3^1) + 1*(3^0)).
we see that in each column there is an equal number of 1's and an equal number of 2's. Moreover, this then implies also that the sums of those two sequences of deficiencies {1, 1, 22} and {1, 4, 19} are equal, as A296074(n) is a function of (can be computed from) a(n).

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A019565, A033879, A117967, A117968, A295882, A296071, A296072, A296074.
Cf. also A293226.
Differs from A305800 for the first time at n=125.

up_to = 65536;
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Rémy Sigrist
A289814(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Rémy Sigrist
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };
A296071(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(A295882(d))))); m; };
A296072(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(A295882(d))))); m; };
Anotsubmitted3(n) = (1/2)*(2 + ((A296071(n)+A296072(n))^2) - A296071(n) - 3*A296072(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted3(n))),"b296073.txt");

Data section extended up to a(125) by Antti Karttunen, Jun 14 2018

A304746 Restricted growth sequence transform of A291760(n), formed from 2-digits in ternary representation of A254103(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 29 2018

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

Cf. A254103, A289814, A291760.

Cf. also A304740, A304748.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291760(n) = A289814(A254103(n));
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; };
v304746 = rgs_transform(vector(65537,n,A291760(n-1)));
A304746(n) = v304746[1+n];

A292246 Base-2 expansion of a(n) encodes the steps where numbers of the form 3k+2 are encountered when map x -> A253889(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 4, 1, 2, 14, 5, 12, 6, 7, 8, 2, 1, 0, 0, 9, 26, 22, 3, 20, 6, 5, 16, 10, 29, 10, 4, 11, 30, 2, 25, 60, 56, 13, 28, 54, 15, 48, 24, 17, 44, 8, 5, 12, 38, 3, 30, 26, 1, 24, 20, 1, 18, 6, 19, 62, 14, 53, 4, 14, 45, 0, 42, 7, 124, 118, 41, 50, 58, 13, 116, 106, 11, 40, 104, 33, 32, 98, 21, 92, 6, 59, 88, 18, 21, 82, 76, 9, 34, 36, 23, 74

Offset: 1

Antti Karttunen, Sep 15 2017

			For n = 2, the starting value is of the form 3k+2, after which follows A253889(3) = 1, the end point of iteration, which is not, thus a(2) = 1*(2^0) = 1.
For n = 4, the starting value is not of the form 3k+2, while A253889(4) = 2 is, thus a(4) = 0*(2^0) + 1*(2^1) = 2.

Cf. A064216, A253889, A254045, A289814, A292243, A292244, A292245.

Cf. also A291759.

f[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; g[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Map[FromDigits[#, 2] &[IntegerDigits[#, 3] /. d_ /; d > 0 :> d - 1] &, Array[a, 96]] (* Michael De Vlieger, Sep 16 2017 *)

a(1) = 0; for n > 1, a(n) = 2*a(A253889(n)) + floor((n mod 3)/2).
a(n) = A289814(A292243(n)).
A000120(a(n)) = A254045(n).
a(n) AND A292244(n) = a(n) AND A292245(n) = 0, where AND is a bitwise-AND (A004198).

A292372 A binary encoding of 2-digits in base-4 representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        1          2           1
   3        0          3           0
   4        0         10           0
   5        0         11           0
   6        1         12           1
   7        0         13           0
   8        2         20          10
   9        2         21          10
  10        3         22          11
  11        2         23          10
  12        0         30           0
  13        0         31           0
  14        1         32           1
  15        0         33           0
  16        0        100           0
  17        0        101           0
  18        1        102           1

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A004198, A007088, A007090, A048724, A059906, A160382, A292370, A292371, A292373.

Cf. A289814 (analogous sequence for base-3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 2, 1, 0], 2], {n, 0, 120}] (* Michael De Vlieger, Sep 21 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==2 else '0' for i in k), 2)
print([a(n) for n in range(121)]) # Indranil Ghosh, Sep 21 2017

def A292372(n): return 0 if (m:=n&~(n<<1)) < 2 else int(bin(m)[-2:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059906(n AND A048724(n)), where AND is a bitwise-AND (A004198).

For all n >= 0, A000120(a(n)) = A160382(n).

A351030 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351031(i) = A351031(j) and A351032(i) = A351032(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 23, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 34, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 46, 36, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of the ordered pair [A351031(n), A351032(n)], or equally, of the ordered pair [A351033(n), A351034(n)].

For all i, j: a(i) = a(j) => A349910(i) = A349910(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A019565, A048673, A289813, A289814, A291759, A304759, A351031, A351032 A351033, A351034.

Cf. also A293226, A349910.

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); };
A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1+factorback(f))/2; };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A291759(n) = A289814(A048673(n));
A304759(n) = A289813(A048673(n));
A351031(n) = { my(m=1); fordiv(n,d,if(dA019565(A304759(d)))); (m); };
A351032(n) = { my(m=1); fordiv(n,d,if(dA019565(A291759(d)))); (m); };
Aux351030(n) = [A351031(n),A351032(n)];
v351030 = rgs_transform(vector(up_to, n, Aux351030(n)));
A351030(n) = v351030[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];

A300222 In ternary (base-3) representation of n, replace 1's with 0's.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2, 0, 0, 2, 6, 6, 8, 18, 18, 20, 18, 18, 20, 24, 24, 26, 0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2, 0, 0, 2, 6, 6, 8, 18, 18, 20, 18, 18, 20, 24, 24, 26, 54, 54, 56, 54, 54, 56, 60, 60, 62, 54, 54, 56, 54, 54, 56, 60, 60, 62, 72, 72, 74, 72, 72, 74, 78, 78, 80, 0, 0, 2, 0, 0, 2, 6, 6, 8, 0, 0, 2

Offset: 0

Antti Karttunen, Mar 14 2018

			For n=46, which in base-3 (A007089) is 1201, replacing 1's with 0's gives 200, and as that is base-3 representation of 18 (= 2*(3^2) + 0*(3^1) + 0*(3^0)), a(46) = 18.

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A004488, A005836, A007089, A244042, A289814.

Cf. A300822 (Moebius transform).

Array[FromDigits[IntegerDigits[#, 3] /. 1 -> 0, 3] &, 93, 0] (* Michael De Vlieger, Mar 17 2018 *)

A244042(n) = fromdigits(apply(x->(x%2), digits(n, 3)), 3);
A300222(n) = (n - A244042(n));
\\ Or directly as:
A300222(n) = fromdigits(apply(x->(if (1==x, 0, x)), digits(n, 3)), 3);

a(n) = n - A244042(n) = 2*A244042(A004488(n)).
a(n) = 2*A005836(1+A289814(n)). [With the current starting offset 1 of A005836.]
a(n) = A300822(n) + A300824(n).

A304748 Restricted growth sequence transform of A291759(n), formed from 2-digits in ternary representation of A048673(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 29 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A048673, A289814, A291759.

Cf. also A304746.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
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; };
v304748 = rgs_transform(vector(65537,n,A291759(n)));
A304748(n) = v304748[n];

A305302 Restricted growth sequence transform of A278222(A291760(n)), constructed from runlengths of 2-digits in base-3 representation of A254103(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 30 2018

For all i, j: A304746(i) = A304746(j) => a(i) = a(j).

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

Cf. A254103, A289814, A291760, A304746.

Cf. also A305301, A305303.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289813
A291760(n) = A289814(A254103(n));
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));
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; };
v305302 = rgs_transform(vector(65538,n,A278222(A291760(n-1))));
A305302(n) = v305302[1+n];

A305303 Restricted growth sequence transform of ordered pair [A278222(A304760(n)), A278222(A291760(n))], constructed from runlengths of 1-digits and 2-digits in base-3 representation of A254103(n).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 30 2018

Restricted growth sequence transform of A290093(A254103(n)).

For all i, j: a(i) = a(j) => A286633(i) = A286633(j) => A286632(i) = A286632(j).

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

Cf. A254103, A289813, A289814, A291760, A304740, A304746, A304760.

Cf. also A286632, A286633, A290093, A290094, A305301, A305302.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
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 A289813
A304760(n) = A289813(A254103(n));
A291760(n) = A289814(A254103(n));
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));
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; };
Aux305303(n) = [A278222(A304760(n)), A278222(A291760(n))];
v305303 = rgs_transform(vector(65538,n,Aux305303(n-1)));
A305303(n) = v305303[1+n];

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