A243071 -id:A243071 - cp's OEIS frontend

A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 5, 6, 9, 5, 4, 7, 4, 6, 10, 5, 11, 5, 6, 8, 5, 5, 12, 9, 7, 6, 13, 6, 14, 7, 5, 10, 15, 6, 5, 5, 8, 8, 16, 5, 6, 7, 9, 11, 17, 6, 18, 12, 6, 6, 7, 7, 19, 9, 10, 6, 20, 6, 21, 13, 5, 10, 6, 8, 22, 7, 5, 14, 23, 7, 8, 15, 11, 8, 24, 6, 7, 11, 12, 16, 9, 7, 25, 6, 7, 6, 26, 9, 27

Offset: 1

Antti Karttunen, Dec 20 2014

nonn

a(n) tells how many iterations of A252463 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A005940 and A163511.

Similarly for A253553 in trees A253563 and A253565. - Antti Karttunen, Apr 14 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the inner lining of the integer partition with Heinz number n, which is also the size of the largest hook of the same partition. For example, the partition with Heinz number 715 is (6,5,3), with diagram
  o o o o o o
  o o o o o
  o o o
which has inner lining
          o o
      o o o
  o o o
and largest hook
  o o o o o o
  o
  o
both of which have size 8, so a(715) = 8.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000040, A000079, A001221, A001222, A005940, A029837, A061395, A156552 (A005941), A163511, A243071, A252461, A252463, A252735, A252736, A252759, A253553, A253563, A253565, A297113, A297155, A297167, A324870, A324872.
Right edge of irregular triangle A265146.
Cf. also A246348.
Cf. A052126, A056239, A093641, A112798, A257990, A325133, A325134, A325135.

Table[If[n==1,1,PrimeOmega[n]+PrimePi[FactorInteger[n][[-1,1]]]]-1,{n,100}] (* Gus Wiseman, Apr 02 2019 *)

A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = (bigomega(n) + A061395(n) - 1); \\ Antti Karttunen, Apr 14 2019

from sympy import primepi, primeomega, primefactors
def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # Chai Wah Wu, Jul 17 2023

a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).
a(n) = A029837(1+A243071(n)). [a(n) = binary width of terms of A243071.]
a(n) = A029837(A005941(n)) = A029837(1+A156552(n)). [Also binary width of terms of A156552.]
Other identities. For all n >= 1:
a(A000040(n)) = n.
a(A001248(n)) = n+1.
a(A030078(n)) = n+2.
And in general, a(prime(n)^k) = n+k-1.
a(A000079(n)) = n. [I.e., a(2^n) = n.]
For all n >= 2:
a(n) = A001222(n) + A061395(n) - 1 = A001222(n) + A252735(n) = A061395(n) + A252736(n) = 1 + A252735(n) + A252736(n).
a(n) = A325134(n) - 1. - Gus Wiseman, Apr 02 2019
From Antti Karttunen, Apr 14 2019: (Start)
a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).
a(n) = A001221(n) + A297167(n) = A297113(n) + A297155(n).
(End).

A245611 Permutation of natural numbers: a(n) = A243071(A064216(n)).

Original entry on oeis.org

0, 1, 3, 7, 2, 15, 31, 6, 63, 127, 14, 255, 5, 4, 511, 1023, 30, 13, 2047, 62, 4095, 8191, 12, 16383, 11, 126, 32767, 29, 254, 65535, 131071, 28, 61, 262143, 510, 524287, 1048575, 10, 27, 2097151, 8, 4194303, 125, 1022, 8388607, 59, 2046, 253, 16777215, 60, 33554431, 67108863, 26

Offset: 1

Antti Karttunen, Jul 28 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

The odd bisection of A243071 decremented by one and halved. (For a(1) = 0, take ceiling of -1/2).

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A245612.

Cf. A054429, A064216, A243071, A243065-A243066, A243505-A243506, A245607, A245609, A244153, A244154.

(define (A245611 n) (A243071 (A064216 n))) ;; offset 1, a(1) = 0.

a(1) = 0, and for n > 1, a(n) = (1/2) * (A243071((2*n)-1) - 1).
As a composition of related permutations:
a(n) = A243071(A064216(n)).
a(n) = A054429(A244153(n)).

A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 14, 32, 24, 10, 15, 64, 13, 128, 28, 20, 48, 256, 30, 9, 96, 11, 56, 512, 26, 1024, 31, 40, 192, 18, 29, 2048, 384, 80, 60, 4096, 52, 8192, 112, 22, 768, 16384, 62, 17, 25, 160, 224, 32768, 27, 36, 120, 320, 1536, 65536, 58, 131072, 3072, 44, 63, 72, 104, 262144, 448, 640, 50, 524288, 61, 1048576, 6144, 21

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Note the indexing: domain starts from one, while the range includes also zero. See also comments in A253564.

The a(n)-th composition in standard order (graded reverse-lexicographic, A066099) is one plus the first differences of the weakly increasing sequence of prime indices of n with 1 prepended. See formula for a simplification. The triangular form is A358169. The inverse is A253565. Not prepending 1 gives A358171. For Heinz numbers instead of standard compositions we have A325351 (without prepending A325352). - Gus Wiseman, Dec 23 2022

			From _Gus Wiseman_, Dec 23 2022: (Start)
This represents the following bijection between partitions and compositions. The reversed prime indices of n together with the a(n)-th composition in standard order are:
   1:        () -> ()
   2:       (1) -> (1)
   3:       (2) -> (2)
   4:     (1,1) -> (1,1)
   5:       (3) -> (3)
   6:     (2,1) -> (1,2)
   7:       (4) -> (4)
   8:   (1,1,1) -> (1,1,1)
   9:     (2,2) -> (2,1)
  10:     (3,1) -> (1,3)
  11:       (5) -> (5)
  12:   (2,1,1) -> (1,1,2)
  13:       (6) -> (6)
  14:     (4,1) -> (1,4)
  15:     (3,2) -> (2,2)
  16: (1,1,1,1) -> (1,1,1,1)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A253565.
Cf. A122111, A243071, A253564, A054429, A336120, A336125.
Applying A000120 gives A001222.
A reverse version is A156552, inverse essentially A005940.
The inverse is A253565, triangular form A242628.
The triangular form is A358169.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A066099 lists standard compositions, lengths A000120, sums A070939.
A112798 list prime indices, sum A056239.
A358134 gives partial sums of standard compositions, Heinz number A358170.
Cf. A029837, A241916, A243055, A287352, A325390, A355534, A358171, A359042.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
stcinv/@Table[Differences[Prepend[primeMS[n],1]]+1,{n,100}] (* Gus Wiseman, Dec 23 2022 *)

(define (A253566 n) (A243071 (A122111 n)))

a(n) = A243071(A122111(n)).
As a composition of other permutations:
a(n) = A054429(A253564(n)).
a(n) = A336120(n) + A336125(n). - Antti Karttunen, Jul 18 2020
If 2n = Product_{i=1..k} prime(x_i) then a(n) = Sum_{i=1..k-1} 2^(x_k-x_{k-i}+i-1). - Gus Wiseman, Dec 23 2022

A332215 Mersenne-prime fixing variant of A243071: a(n) = A243071(A332213(n)).

Original entry on oeis.org

0, 1, 3, 2, 15, 6, 7, 4, 5, 30, 63, 12, 255, 14, 29, 8, 511, 10, 1023, 60, 13, 126, 2047, 24, 23, 510, 9, 28, 4095, 58, 31, 16, 125, 1022, 27, 20, 16383, 2046, 509, 120, 32767, 26, 65535, 252, 57, 4094, 262143, 48, 11, 46, 1021, 1020, 1048575, 18, 119, 56, 2045, 8190, 2097151, 116, 4194303, 62, 25, 32, 503, 250, 8388607, 2044, 4093, 54, 16777215, 40

Offset: 1

Antti Karttunen, Feb 09 2020

nonn

Any Mersenne prime (A000668) times any power of 2 (i.e., 2^k, for k>=0) is fixed by this sequence, including also all even perfect numbers.
From Antti Karttunen, Jul 10 2020: (Start)
This is a "tuned variant" of A243071, and has many of the same properties.
For example, for n > 1, A007814(a(n)) = A007814(n) - A209229(n), that is, this map  preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is decremented by one, and in particular, a(2^k * n) = 2^k * a(n) for all n > 1. Also, like A243071, this bijection maps primes to the terms of A000225 (binary repunits). However, the "tuning" (A332213) has a specific effect that each Mersenne prime (A000668) is mapped to itself. Therefore the terms of A335431 are fixed by this map. Furthermore, I conjecture that there are no other fixed points. For the starters, see the proof in A335879, which shows that at least none of the terms of A335882 are fixed.
(End)

Cf. A243071, A332210, A332213, A332214 (inverse permutation), A335431 (conjectured to be all the fixed points), A335879.

Cf. A000043, A000668, A000396, A324200.

PARI
```
A332215(n) = A243071(A332213(n));
```

a(n) = A243071(A332213(n)).

For all n >= 1, a(A335431(n)) = A335431(n), a(A335882(n)) = A335879(n). - Antti Karttunen, Jul 10 2020

A292943 a(n) = A292944(A243071(n)); Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 4, 0, 1, 4, 8, 4, 16, 8, 5, 0, 32, 2, 64, 8, 9, 16, 128, 8, 2, 32, 1, 16, 256, 10, 512, 0, 17, 64, 10, 4, 1024, 128, 33, 16, 2048, 18, 4096, 32, 9, 256, 8192, 16, 4, 4, 65, 64, 16384, 2, 18, 32, 129, 512, 32768, 20, 65536, 1024, 17, 0, 34, 34, 131072, 128, 257, 20, 262144, 8, 524288, 2048, 5, 256, 20, 66, 1048576, 32, 1, 4096

Offset: 1

Antti Karttunen, Sep 28 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to binary expansion of n

Cf. A005940, A163511, A243071, A292941, A292944, A292945.

Cf. also A292253, A292255, A292383.

(define (A292943 n) (A292944 (A243071 n)))
(define (A292943 n) (if (<= n 1) 0 (+ (if (= 3 (modulo n 6)) 1 0) (* 2 (A292943 (A252463 n))))))

a(n) = A292944(A243071(n)).
a(1) = 0, and for n > 1, a(n) = 2*a(A252463(n)) + [n == 3 (mod 6)], where the last part of the formula is Iverson bracket, giving 1 only if n is of the form 6k+3, and 0 otherwise.
For n >= 0, a(A163511(n)) = A292944(n).
For n >= 1, A292941(n) + a(n) + A292945(n) = a(n) + A292253(n) + A292255(n) = A243071(n).

A364288 a(n) = n - A243071(n).

Original entry on oeis.org

1, 1, 0, 2, -2, 0, -8, 4, 4, -4, -20, 0, -50, -16, 2, 8, -110, 8, -236, -8, -8, -40, -488, 0, 14, -100, 18, -32, -994, 4, -2016, 16, -28, -220, 8, 16, -4058, -472, -86, -16, -8150, -16, -16340, -80, 20, -976, -32720, 0, 26, 28, -202, -200, -65482, 36, -4, -64, -452, -1988, -131012, 8, -262082, -4032, 6, 32, -58, -56

Offset: 1

Antti Karttunen, Jul 25 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..12288

Cf. A243071, A364256 [= gcd(n,a(n))], A364258.
Cf. A007283 (positions of 0's, conjectured), A364289 (positions of terms <= 0), A364290 (of terms > 0), A364291 (of terms >= 0).
Cf. also A364253.

nn = 60; f[x_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First[#], Last[#] ] &@ Transpose@ FactorInteger@ x; Do[a[n] = Which[n <= 2, n - 1, OddQ[n], 1 + 2 a[f[n]], True, 2 a[n/2] ], {n, nn}]; Array[# - a[#] &, nn] (* Michael De Vlieger, Jul 25 2023 *)

A064989(n) = { my(f=factor(n>>valuation(n,2))); 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)))));
A364288(n) = (n-A243071(n));

a(n) = A364258(A243071(n)).
For n >= 1, a(2*n) = 2*a(n).
For n >= 0, a(A007283(n)) = 0.

A292263 a(n) = A292264(A243071(n)).

Original entry on oeis.org

0, 1, 2, 2, 5, 4, 11, 4, 4, 10, 23, 8, 47, 22, 8, 8, 95, 8, 191, 20, 20, 46, 383, 16, 9, 94, 8, 44, 767, 16, 1535, 16, 44, 190, 17, 16, 3071, 382, 92, 40, 6143, 40, 12287, 92, 16, 766, 24575, 32, 19, 18, 188, 188, 49151, 16, 41, 88, 380, 1534, 98303, 32, 196607, 3070, 40, 32, 89, 88, 393215, 380, 764, 34, 786431, 32, 1572863, 6142, 16, 764

Offset: 1

Antti Karttunen, Sep 30 2017

nonn

Base-2 expansion of a(n) encodes the steps where numbers that are neither multiples of 2 nor 3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n. An exception is the most significant bit of a(n) which corresponds with the final 1, but is shifted one bit-position towards right.

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

Cf. A005940, A292253, A292255, A292941, A292945.

a(n) = A292264(A243071(n)).
a(1) = 0, a(2) = 1, and for n > 2, a(n) = 2*a(A252463(n)) + [n = -1 or +1 (mod 6)].
Also, for n > 2, a(n) = 2*a(A252463(n)) + [n == 1 (mod 2)]*[abs(J(3|n)) == 1], where J is the Jacobi-symbol, and [ ]'s are Iverson brackets, whose product gives 1 only if n is an odd number for which J(3|n) = +1 or -1, and 0 otherwise.
a(n) = A292941(n) + A292945(n).
a(n) = A292253(n) + A292255(n).

A297171 Möbius transform of A243071.

Original entry on oeis.org

0, 1, 3, 1, 7, 2, 15, 2, 2, 6, 31, 5, 63, 14, 3, 4, 127, 2, 255, 13, 11, 30, 511, 10, 4, 62, 4, 29, 1023, 4, 2047, 8, 27, 126, 5, 4, 4095, 254, 59, 26, 8191, 12, 16383, 61, 10, 510, 32767, 20, 8, 4, 123, 125, 65535, 4, 21, 58, 251, 1022, 131071, 7, 262143, 2046, 26, 16, 53, 28, 524287, 253, 507, 6, 1048575, 8

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A008683, A064989, A243071, A297161 (rgs-transform of this sequence).

Cf. also A297112, A297156, A297172.

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)))));
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));

a(n) = Sum_{d|n} A008683(n/d)*A243071(d).

A364497 Numbers k such that k divides A243071(k).

Original entry on oeis.org

1, 3, 6, 12, 24, 43, 48, 86, 96, 172, 192, 344, 384, 688, 768, 1177, 1376, 1536, 2354, 2752, 3072, 3503, 4708, 5504, 6144, 7006, 9416, 11008, 12288, 14012, 18832, 22016, 24576, 28024, 37664, 44032, 49152, 49477, 56048, 75328, 88064, 98304, 98954, 112096, 150656, 169413, 176128, 196608, 197908, 224192, 301312, 338826

Offset: 1

Antti Karttunen, Jul 27 2023

nonn

If n is present, then 2*n is also present, and vice versa.
A007283 is included as a subsequence, because it gives the known fixed points of map n -> A163511(n).
Sequence A163511(A364496(.)) sorted into ascending order.

Index entries for sequences computed from indices in prime factorization

Cf. A163511, A243071.
Cf. A007283 (subsequence), A364498 (odd terms).
Cf. also A364295, A364494, A364496.

A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A243071(n) = if(n<=2, n-1, A054429(A156552(n)));
isA364497(n) = !(A243071(n)%n);

A364289 Numbers k such that A243071(k) >= k.

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 31, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 104, 106, 107, 109

Offset: 1

Antti Karttunen, Jul 25 2023

nonn

If k is present, then 2*k is also present, and vice versa.

Positions of terms <= 0 in A364288.
Cf. A007283 (subsequence), A243071, A364290 (complement).
Cf. also A364287.

A064989(n) = { my(f=factor(n>>valuation(n,2))); 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)))));
isA364289(n) = (A243071(n)>=n);

cp's OEIS Frontend

A252464 a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A245611 Permutation of natural numbers: a(n) = A243071(A064216(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A332215 Mersenne-prime fixing variant of A243071: a(n) = A243071(A332213(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A292943 a(n) = A292944(A243071(n)); Base-2 expansion of a(n) encodes the steps where numbers of the form 6k+3 are encountered when map x -> A252463(x) is iterated down to 1, starting from x=n.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula

A364288 a(n) = n - A243071(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A292263 a(n) = A292264(A243071(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A297171 Möbius transform of A243071.

Views

Author

Keywords

Links

Crossrefs

Programs