A278220 -id:A278220 - cp's OEIS frontend

A331280 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278220(i) = A278220(j) for all i, j.

1, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 9, 8, 6, 5, 14, 9, 15, 2, 16, 10, 17, 6, 18, 11, 19, 4, 20, 12, 21, 7, 9, 13, 22, 3, 12, 9, 23, 8, 24, 6, 25, 5, 26, 14, 27, 9, 28, 15, 12, 2, 29, 16, 30, 10, 31, 17, 32, 6, 33, 18, 34, 11, 25, 19, 35, 4, 6, 20, 36, 12, 37, 21, 38, 7, 39, 9, 40, 13, 41, 22, 42, 3, 43, 12, 16, 9, 44, 23, 45, 8, 46

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Restricted growth sequence transform of A278220(n) (= A046523(A241909(n))).

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

Cf. A046523, A241909, A278220.

Cf. also A286621, A331299.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from A046523
A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));
A278220(n) = A046523(A241909(n));
v331280 = rgs_transform(vector(up_to, n, A278220(n)));
A331280(n) = v331280[n];

A241909 Self-inverse permutation of natural numbers: a(1)=1, a(p_i) = 2^i, and if n = p_i1 * p_i2 * p_i3 * ... * p_{ik-1} * p_ik, where p's are primes, with their indexes are sorted into nondescending order: i1 <= i2 <= i3 <= ... <= i_{k-1} <= ik, then a(n) = 2^(i1-1) * 3^(i2-i1) * 5^(i3-i2) * ... * p_k^(1+(ik-i_{k-1})). Here k = A001222(n) and ik = A061395(n).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 16, 5, 6, 27, 32, 25, 64, 81, 18, 7, 128, 15, 256, 125, 54, 243, 512, 49, 12, 729, 10, 625, 1024, 75, 2048, 11, 162, 2187, 36, 35, 4096, 6561, 486, 343, 8192, 375, 16384, 3125, 50, 19683, 32768, 121, 24, 45, 1458, 15625, 65536, 21, 108, 2401

Offset: 1

Antti Karttunen, May 03 2014, partly inspired by Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

nonn

This permutation maps between the partitions as ordered in A112798 and A241918 (the original motivation for this sequence).
For all n > 2, A007814(a(n)) = A055396(n)-1, which implies that this self-inverse permutation maps between primes (A000040) and the powers of two larger than one (A000079(n>=1)), and apart from a(1) & a(2), this also maps each even number to some odd number, and vice versa, which means there are no fixed points after 2.
A122111 commutes with this one, that is, a(n) = A122111(a(A122111(n))).
Conjugates between A243051 and A242424 and other rows of A243060 and A243070.

			For n = 12 = 2 * 2 * 3 = p_1 * p_1 * p_2, we obtain by the first formula 2^(1-1) * 3^(1-1) * 5^(1+(2-1)) = 5^2 = 25. By the second formula, as n = 2^2 * 3^1, we obtain the same result, p_{1+2} * p_{2+1} = p_3 * p_3 = 25, thus a(12) = 25.
Using the product formula over the terms of row n of table A241918, we see, because 9450 = 2*3*3*3*5*5*7 = p_1^1 * p_2^3 * p_3^2 * p_4^1, that the corresponding row in A241918 is {2,5,7,7}, and multiplying p_2 * p_5 * p_7^2 yields 3 * 11 * 17 * 17 = 9537, thus a(9450) = 9537.
Similarly, for 9537, the corresponding row in A241918 is {1,2,2,2,3,3,4}, and multiplying p_1^1 * p_2^3 * p_3^2 * p_4^1, we obtain 9450 back.

Antti Karttunen, Table of n, a(n) for n = 1..1024
A. Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

Cf. A203623, A241918, A242378, A007814, A064989, A064216, A001222, A061395, A125976, A243060, A243070.
Cf. also A278220 (= A046523(a(n))), A331280 (its rgs_transform), A331299 (= min(n,a(n))).
{A000027, A122111, A241909, A241916} form a 4-group.
Cf. A049084, A027746.

a241909 1 = 1
a241909 n = product $ zipWith (^) a000040_list $ zipWith (-) is (1 : is)
            where is = reverse ((j + 1) : js)
                  (j:js) = reverse $ map a049084 $ a027746_row n
-- Reinhard Zumkeller, Aug 04 2014

Array[If[# == 1, 1, Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[FactorInteger[#] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]] &, 56] (* Michael De Vlieger, Jan 23 2020 *)

A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m)); \\ Antti Karttunen, Jan 17 2020

If n is a prime with index i (p_i), then a(n) = 2^i, otherwise when n = p_i1 * p_i2 * p_i3 * ... p_ik, where p_i1, p_i2, p_i3, ..., p_ik are the primes present (not necessarily all distinct) in the prime factorization of n, sorted into nondescending order, a(n) = 2^(i1-1) * 3^(i2-i1) * 5^(i3-i2) * ... * p_k^(1+(ik-i_{k-1})).
Equally, if n = 2^k, then a(n) = p_k, otherwise, when n = 2^e1 * 3^e2 * 5^e3 * ... * p_k^{e_k}, i.e., where e1 ... e_k are the exponents (some of them possibly zero, except the last) of the primes 2, 3, 5, ... in the prime factorization of n, a(n) = p_{1+e1} * p_{1+e1+e2} * p_{1+e1+e2+e3} * ... * p_{e1+e2+e3+...+e_k}.
From the equivalence of the above two formulas (which are inverses of each other) it follows that a(a(n)) = n, i.e., that this permutation is an involution. For a proof, please see the attached notes.
The first formula corresponds to this recurrence:
a(1) = 1, a(p_k) = 2^k for primes with index k, otherwise a(n) = (A000040(A001222(n))^(A241917(n)+1)) * A052126(a(A052126(n))).
And the latter formula with this recurrence:
a(1) = 1, and for n>1, if n = 2^k, a(n) = A000040(k), otherwise a(n) = A000040(A001511(n)) * A242378(A007814(n), a(A064989(n))).
[Here A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n.]
We also have:
a(1)=1, and for n>1, a(n) = Product_{i=A203623(n-1)+2..A203623(n)+1} A000040(A241918(i)).
For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).
For all n > 1, a(2n-1) = 2*a(A064216(n)).

Typos in the name corrected by Antti Karttunen, May 31 2014

A278221 Filtering sequence (related to prime factorization): a(n) = A046523(A122111(n)).

Original entry on oeis.org

1, 2, 4, 2, 8, 6, 16, 2, 4, 12, 32, 6, 64, 24, 12, 2, 128, 6, 256, 12, 36, 48, 512, 6, 8, 96, 4, 24, 1024, 30, 2048, 2, 72, 192, 24, 6, 4096, 384, 144, 12, 8192, 60, 16384, 48, 12, 768, 32768, 6, 16, 12, 288, 96, 65536, 6, 72, 24, 576, 1536, 131072, 30, 262144, 3072, 36, 2, 216, 120, 524288, 192, 1152, 60, 1048576, 6, 2097152, 6144, 12, 384, 48, 240, 4194304, 12, 4

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A046523, A122111.

Cf. also A278220.

(define (A278221 n) (A046523 (A122111 n)))

a(n) = A046523(A122111(n)).

A278219 Filter-sequence related to base-2 run-length encoding: a(n) = A046523(A243353(n)).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 6, 2, 4, 12, 16, 8, 6, 12, 6, 2, 4, 12, 36, 12, 16, 32, 24, 8, 6, 30, 24, 12, 6, 12, 6, 2, 4, 12, 36, 12, 36, 72, 60, 12, 16, 48, 64, 32, 24, 72, 24, 8, 6, 30, 60, 30, 24, 48, 60, 12, 6, 30, 24, 12, 6, 12, 6, 2, 4, 12, 36, 12, 36, 72, 60, 12, 36, 180, 144, 72, 60, 180, 60, 12, 16, 48, 144, 48, 64, 128, 96, 32, 24, 120, 216, 72, 24, 72

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

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

Cf. A003188, A046523, A075157, A243353, A278220.
Other base-2 related filter sequences: A278217, A278222.
Sequences that (seem to) partition N into same or coarser equivalence classes are at least these: A005811, A136004, A033264, A037800, A069010, A087116, A090079 and many others like A105500, A106826, A166242, A246960, A277561, A037834, A225081 although these have not been fully checked yet.

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]]];
Table[g@ f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, 93}] (* Michael De Vlieger, May 09 2017 *)

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 a003188(n): return n^int(n/2)
def a243353(n): return a005940(1 + a003188(n))
def a(n): return a046523(a243353(n)) # Indranil Ghosh, May 07 2017

(define (A278219 n) (A046523 (A243353 n)))

a(n) = A046523(A243353(n)).
a(n) = A278222(A003188(n)).
a(n) = A278220(1+A075157(n)).

A279354 a(n) = A046523(A279356(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, 6, 32, 512, 4, 12, 64, 24, 16, 1024, 12, 2048, 2, 12, 128, 36, 6, 4096, 256, 48, 8, 8192, 24, 16384, 32, 6, 512, 32768, 4, 24, 12, 30, 64, 65536, 6, 12, 16, 24, 1024, 131072, 12, 262144, 2048, 96, 2, 72, 48, 524288, 128, 12, 36, 1048576, 6, 2097152, 4096, 192, 256, 72, 96, 4194304, 8, 48, 8192

Offset: 1

Antti Karttunen, Dec 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..818

Cf. A046523, A249818, A278220, A278523, A279350, A279356.

a(n) = A046523(A279356(n)).

a(n) = A278220(A249818(n)).

A278525 Filtering sequence (related to prime factorization): a(n) = A046523(A241916(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 8, 6, 4, 2, 8, 2, 4, 6, 16, 2, 12, 2, 8, 6, 4, 2, 16, 6, 4, 12, 8, 2, 12, 2, 32, 6, 4, 6, 24, 2, 4, 6, 16, 2, 12, 2, 8, 12, 4, 2, 32, 6, 12, 6, 8, 2, 36, 6, 16, 6, 4, 2, 24, 2, 4, 12, 64, 6, 12, 2, 8, 6, 12, 2, 48, 2, 4, 30, 8, 6, 12, 2, 32, 24, 4, 2, 24, 6, 4, 6, 16, 2, 36, 6, 8, 6, 4, 6, 64, 2, 12, 12, 24, 2, 12, 2, 16, 30, 4, 2, 72, 2

Offset: 1

Antti Karttunen, Nov 30 2016

nonn

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

Cf. A000040, A046523, A241916, A278220, A278221.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A209229(n) = (n && !bitand(n,n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf,1]), g = f); for(i=1,nbf,g[i,1] = prime(1+igp-primepi(f[i,1]))); factorback(g)/2);
A278525(n) = A046523(A241916(n)); \\ Antti Karttunen, Jul 02 2018

(define (A278525 n) (A046523 (A241916 n)))

a(n) = A046523(A241916(n)).
Other identities. For all n:
a(2^n) = 2^n.
a(A000040(n)) = 2.

cp's OEIS Frontend

A331280 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278220(i) = A278220(j) for all i, j.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A278221 Filtering sequence (related to prime factorization): a(n) = A046523(A122111(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A278219 Filter-sequence related to base-2 run-length encoding: a(n) = A046523(A243353(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

A279354 a(n) = A046523(A279356(n)).

Views

Author

Keywords

Links

Crossrefs

Formula

A278525 Filtering sequence (related to prime factorization): a(n) = A046523(A241916(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Scheme

Formula