A278219 -id:A278219 - cp's OEIS frontend

A286619 Restricted growth sequence computed for filter-sequence A278219, related to run-lengths in the binary representation of n.

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

Offset: 0

Antti Karttunen, May 11 2017

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A278219, because for all i, j it holds that: a(i) = a(j) <=> A278219(i) = A278219(j).

For example, for all i, j: a(i) = a(j) => A005811(i) = A005811(j). (The same is true for A073334, as it is a sequence computed from A005811).

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

Cf. A278219, A005811, A073334.

Cf. also A101296, A286603, A286605, A286610, A286621, A286622, A286626, A286378 for similarly constructed sequences.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; With[{nn = 99}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[g@ f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, nn}]] (* Michael De Vlieger, May 12 2017, Version 10 *)

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A003188(n) = bitxor(n, n>>1);
A278219(n) = A278222(A003188(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278219(n-1))),"b286619.txt");

A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 2, 6, 7, 8, 4, 5, 9, 10, 2, 3, 7, 11, 7, 8, 12, 13, 4, 14, 15, 16, 9, 17, 18, 14, 2, 6, 7, 19, 20, 11, 21, 22, 7, 23, 24, 25, 26, 27, 28, 27, 4, 14, 29, 30, 15, 31, 32, 33, 9, 34, 35, 36, 9, 37, 38, 37, 2, 39, 40, 11, 7, 41, 42, 43, 40, 41, 44, 45, 46, 47, 48, 47, 7, 49, 50, 51, 24, 52, 53, 54, 55, 56, 57, 58, 42, 59, 60, 56, 4, 61, 62, 63

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

Restricted growth sequence transform of the ordered pair [A278219(n), A324386(n)].

Cf. A278219, A324386.

Cf. also A286619, A324343, A324344, A324380 (compare the scatter-plots).

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 }; \\ From A005940
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));
A003188(n) = bitxor(n, n>>1);
A278219(n) = A278222(A003188(n));
Aux324390(n) = [A278219(n), A324386(n)]; \\ See code for A324386 in that entry.
v324390 = rgs_transform(vector(1+up_to,n,Aux324390(n-1)));
A324390(n) = v324390[1+n];

a(A000225(n)) = 2 for all n >= 1.

A286242 Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 12, 14, 12, 84, 40, 44, 12, 142, 216, 183, 40, 265, 86, 152, 12, 142, 826, 265, 216, 1860, 607, 489, 40, 832, 607, 1117, 86, 619, 226, 560, 12, 142, 826, 265, 826, 5080, 2497, 619, 216, 2956, 4308, 4155, 607, 8575, 1105, 1533, 40, 832, 2497, 2116, 607, 5731, 4501, 3475, 86, 1402, 1105, 3475, 226, 1759, 698, 2144, 12, 142, 826, 265, 826, 5080, 2497, 619, 826

Offset: 0

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
MathWorld, Pairing Function

Cf. A000027, A003188, A278219, A278222, A286240, A286241, A286242.

A003188(n) = bitxor(n, n>>1);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A278219(n) = A278222(A003188(n));
A286242(n) = (2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n))/2;
for(n=0, 16383, write("b286242.txt", n, " ", A286242(n)));

from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f=factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a003188(n): return n^(n>>1)
def a243353(n): return a005940(1 + a003188(n))
def a278219(n): return a046523(a243353(n))
def a278222(n): return a046523(a005940(n + 1))
def a(n): return T(a278222(n), a278219(n)) # Indranil Ghosh, May 07 2017

(define (A286242 n) (* (/ 1 2) (+ (expt (+ (A278222 n) (A278219 n)) 2) (- (A278222 n)) (- (* 3 (A278219 n))) 2)))

a(n) = (1/2)*(2 + ((A278222(n)+A278219(n))^2) - A278222(n) - 3*A278219(n)).

a(n) = (1/2)*(2 + ((A278222(n)+A278222(A003188(n)))^2) - A278222(n) - 3*A278222(A003188(n))).

A286554 Ordinal transform of A286619, or equally, of A278219.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 21 2017

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

Cf. A278219, A286619, A286552, A286558.

A352888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278219(A109812(i)) = A278219(A109812(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 07 2022

Restricted growth sequence transform of A278219(A109812(n)).

For all i, j: a(i) = a(j) => A352889(i) = A352889(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to binary expansion of n

Cf. A109812, A278219, A352889.

Cf. also A286619, A351578, A351963.

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; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
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
A003188(n) = bitxor(n, n>>1);
A278219(n) = A046523(A005940(1+A003188(n)));
A109812(n) = v109812[n];
v352888 = rgs_transform(vector(up_to, n, A278219(A109812(n))));
A352888(n) = v352888[n];

A286241 Compound filter: a(n) = P(A278219(n), A278219(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 12, 14, 12, 59, 86, 27, 12, 109, 363, 269, 86, 142, 148, 27, 12, 109, 1093, 1117, 363, 1097, 1517, 489, 86, 601, 1408, 619, 148, 142, 148, 27, 12, 109, 1093, 1117, 1093, 5707, 8587, 2545, 363, 1969, 6153, 4529, 1517, 4489, 4537, 489, 86, 601, 3946, 3976, 1408, 2509, 5719, 2545, 148, 601, 1408, 619, 148, 142, 148, 27, 12, 109, 1093, 1117, 1093, 5707, 8587

Offset: 0

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A003188, A005940, A046523, A243353, A278219, A278222, A286240, A286242, A286255.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; h[n_] := g@ f[BitXor[n, Floor[n/2]], 1, 1]; Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ # &, Table[{h[n], h[n + 1]}, {k, 12}, {n, k (k - 1)/2, k (k + 1)/2 - 1}]] // Flatten (* Michael De Vlieger, May 09 2017 *)

A003188(n) = bitxor(n, n>>1);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278222(n) = A046523(A005940(1+n));
A278219(n) = A278222(A003188(n));
A286241(n) = (2 + ((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n))/2;
for(n=0, 16383, write("b286241.txt", n, " ", A286241(n)));

from sympy import prime, factorint
import math
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n))
def a278219(n): return a046523(a243353(n))
def a(n): return T(a278219(n), a278219(n + 1)) # Indranil Ghosh, May 07 2017

(define (A286241 n) (* (/ 1 2) (+ (expt (+ (A278219 n) (A278219 (+ 1 n))) 2) (- (A278219 n)) (- (* 3 (A278219 (+ 1 n)))) 2)))

a(n) = (1/2)*(2+((A278219(n)+A278219(1+n))^2) - A278219(n) - 3*A278219(1+n)).

A278532 a(n) = A278219(A255056(n)).

Original entry on oeis.org

1, 4, 4, 6, 16, 6, 6, 36, 24, 24, 6, 6, 4, 36, 64, 24, 60, 60, 24, 6, 6, 4, 36, 144, 60, 64, 216, 24, 6, 60, 96, 60, 60, 60, 24, 6, 6, 4, 36, 144, 60, 144, 60, 60, 144, 64, 96, 216, 216, 24, 6, 60, 240, 210, 96, 360, 60, 6, 60, 96, 60, 60, 60, 24, 6, 6, 4, 36, 144, 60, 144, 60, 60, 900, 144, 360, 360, 60, 60, 144, 144, 240, 384, 96, 360, 216, 360, 216, 216, 24

Offset: 0

Antti Karttunen, Nov 30 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16142

Cf. A255056, A255336, A278219, A278232, A278262, A278534.

(define (A278532 n) (A278219 (A255056 n)))

a(n) = A278219(A255056(n)).

A324533 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278219(i) = A278219(j), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 05 2019

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A278219(n)].

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

Cf. A002487, A278219, A286619, A324400.

Cf. also A323889 (compare the scatterplots).

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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
A003188(n) = bitxor(n, n>>1);
A278219(n) = A046523(A005940(1+A003188(n)));
Aux324533(n) = [A002487(n), A278219(n)];
v324533 = rgs_transform(vector(1+up_to,n,Aux324533(n-1)));
A324533(n) = v324533[1+n];

For n >= 1, a(2^n) = 3.

A278222 The least number with the same prime signature as A005940(n+1).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 6, 6, 12, 4, 12, 8, 16, 2, 6, 6, 12, 6, 30, 12, 24, 4, 12, 12, 36, 8, 24, 16, 32, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 4, 12, 12, 36, 12, 60, 36, 72, 8, 24, 24, 72, 16, 48, 32, 64, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 6, 30, 30, 60, 30, 210, 60, 120, 12, 60, 60, 180, 24, 120, 48, 96, 4, 12, 12

Offset: 0

Antti Karttunen, Nov 15 2016

This sequence can be used for filtering certain base-2 related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A005940(n+1)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Because the Doudna map n -> A005940(1+n) is an isomorphism from "unary-binary encoding of factorization" (see A156552) to the ordinary representation of the prime factorization of n, it follows that the equivalence classes of this sequence match with any such sequence b, where b(n) is computed from the lengths of 1-runs in the binary representation of n and the order of those 1-runs does not matter. Particularly, this holds for any sequence that is obtained as a "Run Length Transform", i.e., where b(n) = Product S(i), for some function S, where i runs through the lengths of runs of 1's in the binary expansion of n. See for example A227349.
However, this sequence itself is not a run length transform of any sequence (which can be seen for example from the fact that A046523 is not multiplicative).
Furthermore, this matches not only with sequences involving products of S(i), but with any sequence obtained with any commutative function applied cumulatively, like e.g., A000120 (binary weight, obtained in this case as Sum identity(i)), and A069010 (number of runs of 1's in binary representation of n, obtained as Sum signum(i)).

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

Cf. A005940, A156552, A006068, A124859, A278159.
Similar sequences: A278217, A278219 (other base-2 related variants), A069877 (base-10 related), A278226 (primorial base), A278234-A278236 (factorial base), A278243 (Stern polynomials), A278233 (factorization in ring GF(2)[X]), A046523 (factorization in Z).
Cf. also A286622 (rgs-transform of this sequence) and A286162, A286252, A286163, A286240, A286242, A286379, A286464, A286374, A286375, A286376, A286243, A286553 (various other sequences involving this sequence).
Sequences that partition N into same or coarser equivalence classes: too many to list all here (over a hundred). At least every sequence listed under index-entry "Run Length Transforms" is included (e.g., A227349, A246660, A278159), and also sequences like A000120 and A069010, and their combinations like A136277.

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Array[If[# == 1, 1, Times @@ MapIndexed[ Prime[First[#2]]^#1 &, Sort[FactorInteger[#][[All, -1]], Greater]]] &@ f[# - 1, 1, 1] &, 99] (* Michael De Vlieger, May 09 2017 *)

A046523(n)=factorback(primes(#n=vecsort(factor(n)[, 2], , 4)), n)
a(n)=my(p=2, t=1); for(i=0,exponent(n), if(bittest(n,i), t*=p, p=nextprime(p+1))); A046523(t) \\ Charles R Greathouse IV, Nov 11 2021

from sympy import prime, factorint
import math
def A(n): return n - 2**int(math.floor(math.log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a005940(n): return b(n - 1)
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return a046523(a005940(n + 1)) # Indranil Ghosh, May 05 2017

(define (A278222 n) (A046523 (A005940 (+ 1 n))))

a(n) = A046523(A005940(1+n)).
a(n) = A124859(A278159(n)).
a(n) = A278219(A006068(n)).

Misleading part of the name removed by Antti Karttunen, Apr 07 2022

A278220 Filtering sequence (related to prime factorization): a(n) = A046523(A241909(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 4, 16, 2, 6, 8, 32, 4, 64, 16, 12, 2, 128, 6, 256, 8, 24, 32, 512, 4, 12, 64, 6, 16, 1024, 12, 2048, 2, 48, 128, 36, 6, 4096, 256, 96, 8, 8192, 24, 16384, 32, 12, 512, 32768, 4, 24, 12, 192, 64, 65536, 6, 72, 16, 384, 1024, 131072, 12, 262144, 2048, 24, 2, 144, 48, 524288, 128, 768, 36, 1048576, 6, 2097152, 4096, 30, 256, 72, 96, 4194304, 8, 6, 8192

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..410

Cf. A046523, A075158, A241909.

Cf. also A278219, A278221.

(define (A278220 n) (A046523 (A241909 n)))

a(n) = A046523(A241909(n)).

a(n) = A278219(A075158(n-1)).

cp's OEIS Frontend

A286619 Restricted growth sequence computed for filter-sequence A278219, related to run-lengths in the binary representation of n.

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A324390 Lexicographically earliest positive sequence such that a(i) = a(j) => A278219(i) = A278219(j) and A324386(i) = A324386(j), for all i, j >= 0.

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A286242 Compound filter: a(n) = P(A278222(n), A278219(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A286554 Ordinal transform of A286619, or equally, of A278219.

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A352888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278219(A109812(i)) = A278219(A109812(j)), for all i, j >= 1.

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A286241 Compound filter: a(n) = P(A278219(n), A278219(1+n)), where P(n,k) is sequence A000027 used as a pairing function.

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A278532 a(n) = A278219(A255056(n)).

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A324533 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278219(i) = A278219(j), for all i, j >= 0.

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A278222 The least number with the same prime signature as A005940(n+1).

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