A246346 -id:A246346 - cp's OEIS frontend

A135141 a(1)=1, a(p_n)=2a(n), a(c_n)=2a(n)+1, where p_n = n-th prime, c_n = n-th composite number.

1, 2, 4, 3, 8, 5, 6, 9, 7, 17, 16, 11, 10, 13, 19, 15, 12, 35, 18, 33, 23, 21, 14, 27, 39, 31, 25, 71, 34, 37, 32, 67, 47, 43, 29, 55, 22, 79, 63, 51, 20, 143, 26, 69, 75, 65, 38, 135, 95, 87, 59, 111, 30, 45, 159, 127, 103, 41, 24, 287, 70, 53, 139, 151, 131, 77, 36, 271, 191

Offset: 1

Katarzyna Matylla, Feb 13 2008

A permutation of the positive integers, related to A078442.
a(p) is even when p is prime and is divisible by 2^(prime order of p).
From Robert G. Wilson v, Feb 16 2008: (Start)
What is the length of the cycle containing 10? Is it infinite? The cycle begins 10, 17, 12, 11, 16, 15, 19, 18, 35, 29, 34, 43, 26, 31, 32, 67, 36, 55, 159, 1055, 441, 563, 100, 447, 7935, 274726911, 1013992070762272391167, ... Implementation in Mmca: NestList[a(AT)# &, 10, 26] Furthermore, it appears that any non-single-digit number has an infinite cycle.
Records: 1, 2, 4, 8, 9, 17, 19, 35, 39, 71, 79, 143, 159, 287, 319, 575, 639, 1151, 1279, 2303, 2559, 4607, 5119, 9215, 10239, 18431, 20479, 36863, 40959, 73727, 81919, 147455, 163839, 294911, 327679, 589823, 655359, ..., . (End)

			a(20) = 33 = 2*16 + 1 because 20 is 11th composite and a(11)=16. Or, a(20)=33=100001(bin). In other words it is a composite number, its index is a prime number, whose index is a prime....

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences computed from indices in prime factorization
Index entries for sequences that are permutations of the natural numbers

Cf. A000720, A007814, A010051, A026238, A078442.
Cf. A246346, A246347 (record positions and values).
Cf. A227413 (inverse).
Cf. A071574, A245701, A245702, A245703, A245704, A246377, A236854, A237427 for related and similar permutations.

import Data.List (genericIndex)
a135141 n = genericIndex a135141_list (n-1)
a135141_list = 1 : map f [2..] where
   f x | iprime == 0 = 2 * (a135141 $ a066246 x) + 1
       | otherwise   = 2 * (a135141 iprime)
       where iprime = a049084 x
-- Reinhard Zumkeller, Jan 29 2014

a[1] = 1; a[n_] := If[PrimeQ@n, 2*a[PrimePi[n]], 2*a[n - 1 - PrimePi@n] + 1]; Array[a, 69] (* Robert G. Wilson v, Feb 16 2008 *)

/* Let pc = prime count (which prime it is), cc = composite count: */
pc[1]:0;
cc[1]:0;
pc[2]:1;
cc[4]:1;
pc[n]:=if primep(n) then 1+pc[prev_prime(n)] else 0;
cc[n]:=if primep(n) then 0 else if primep(n-1) then 1+cc[n-2] else 1+cc[n-1];
a[1]:1;
a[n]:=if primep(n) then 2*a[pc[n]] else 1+2*a[cc[n]];

A135141(n) = if(1==n, 1, if(isprime(n), 2*A135141(primepi(n)), 1+(2*A135141(n-primepi(n)-1)))); \\ Antti Karttunen, Dec 09 2019

from sympy import isprime, primepi
def a(n): return 1 if n==1 else 2*a(primepi(n)) if isprime(n) else 2*a(n - 1 - primepi(n)) + 1 # Indranil Ghosh, Jun 11 2017, after Mathematica code

a(n) = 2*A135141((A049084(n))*chip + A066246(n)*(1-chip)) + 1 - chip, where chip = A010051(n). - Reinhard Zumkeller, Jan 29 2014
From Antti Karttunen, Dec 09 2019: (Start)
A007814(a(n)) = A078442(n).
A070939(a(n)) = A246348(n).
A080791(a(n)) = A246370(n).
A054429(a(n)) = A246377(n).
A245702(a(n)) = A245703(n).
a(A245704(n)) = A245701(n). (End)

A246348 a(1)=1, a(p_n) = 1 + a(n), a(c_n) = 1 + a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also binary width of terms of A135141.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 4, 3, 5, 5, 4, 4, 4, 5, 4, 4, 6, 5, 6, 5, 5, 4, 5, 6, 5, 5, 7, 6, 6, 6, 7, 6, 6, 5, 6, 5, 7, 6, 6, 5, 8, 5, 7, 7, 7, 6, 8, 7, 7, 6, 7, 5, 6, 8, 7, 7, 6, 5, 9, 7, 6, 8, 8, 8, 7, 6, 9, 8, 8, 7, 7, 6, 8, 6, 7, 9, 8, 6, 8, 7, 6, 5, 10, 8, 7, 9, 9, 6, 9, 8, 7, 10

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

If n = 1, the result is 1, otherwise, if n is prime, compute the result for that prime's index (A000720 or A049084) and add one, and if n is composite, compute the result for that composite's index (A065855) and add one.

a(n) tells how many calls (including the toplevel call) are required to compute A135141(n) or A246377(n) with a simple (nonmemoized) recursive algorithm as employed for example by Robert G. Wilson v's Mathematica-program of Feb 16 2008 in A135141 or Antti Karttunen's Scheme-proram in A246377.

Antti Karttunen, Table of n, a(n) for n = 1..32998

Cf. A000040, A002808, A000720, A065855, A070939, A135141, A246346, A246347, A246369, A246370, A246377.

\\ Compute the b-files for both the positions of records (A246346) and their values (A246347) and also for A246348 (somewhat naively):
default(primelimit, (2^31)+(2^30));
A070939 = n->#binary(n)+!n \\ From M. F. Hasler
A135141(n) = if(1==n, 1, if(isprime(n), 2*A135141(primepi(n)), 1+(2*A135141(n-primepi(n)-1))));
A246348(n) = A070939(A135141(n));
prevmax = -1; i = 0; for(n=1, 123456, if((k=A135141(n)) > prevmax, prevmax = k; i++; write("b246346.txt", i, " ", n); write("b246347.txt", i, " ", k)); write("b246348.txt", n, " ", A246348(n)));
(Scheme, two versions, second being a direct recurrence employing memoizing definec-macro from Antti Karttunen's IntSeq-library)
(define (A246348 n) (A070939 (A135141 n)))
(definec (A246348 n) (cond ((= 1 n) 1) ((= 1 (A010051 n)) (+ 1 (A246348 (A000720 n)))) (else (+ 1 (A246348 (A065855 n))))))

a(1) = 1, and for n >= 1, if A010051(n)=1 [that is, when n is prime], a(n) = 1 + a(A000720(n)), otherwise a(n) = 1 + a(A065855(n)). [A000720(n) and A065855(n) tell the number of primes, and respectively, composites <= n].
a(n) = A246369(n) + A246370(n).
a(n) = A070939(A135141(n)) = 1 + floor(log_2(A135141(n))). [Sequence gives also the binary width of terms of A135141].
a(n) = A070939(A246377(n)). [Also for 0/1-swapped version of that sequence].

A246347 Record values in A135141.

Original entry on oeis.org

1, 2, 4, 8, 9, 17, 19, 35, 39, 71, 79, 143, 159, 287, 319, 575, 639, 1151, 1279, 2303, 2559, 4607, 5119, 9215, 10239, 18431, 20479, 36863, 40959, 73727, 81919, 147455, 163839, 294911, 327679, 589823, 655359, 1179647, 1310719, 2359295, 2621439, 4718591, 5242879, 9437183, 10485759

Offset: 1

Antti Karttunen, Aug 27 2014 after Robert G. Wilson v's note in A135141

nonn

In binary, the terms of the sequence seem to follow a simple pattern:
          1 = a(1)
         10 = a(2)
        100 = a(3)
       1000 = a(4)
       1001 = a(5)
      10001 = a(6)
      10011 = a(7)
     100011 = a(8)
     100111 = a(9)
    1000111 = a(10)
    1001111 = a(11)
   10001111 = a(12)
   10011111 = a(13)
  100011111 = a(14)
  100111111 = a(15)
  ...
thus the sequence seems to consist of, after 1 and 2, an interleaving of sequence A153894: 4, 9, 19, 39, 79, 159, 319, ... with the sequence A052996 from its third term onward: 8, 17, 35, 71, 143, ...

Antti Karttunen, Table of n, a(n) for n = 1..112

A246346 gives the corresponding positions in A135141.

Cf. A052996, A153894, A246348, A246360.

Union[FromDigits[#,2]&/@Flatten[Table[{PadRight[{1,0,0},n,1],PadRight[ {1,0,0,0},n,1]},{n,30}],1]] (* Harvey P. Dale, May 03 2015 *)

PARI
```
\\ See code in A246348.
```

(define (A246347 n) (A135141 (A246346 n)))

a(n) = A135141(A246346(n)).

cp's OEIS Frontend

A135141 a(1)=1, a(p_n)=2a(n), a(c_n)=2a(n)+1, where p_n = n-th prime, c_n = n-th composite number.

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A246348 a(1)=1, a(p_n) = 1 + a(n), a(c_n) = 1 + a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also binary width of terms of A135141.

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A246347 Record values in A135141.

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cp's OEIS Frontend

A135141 a(1)=1, a(p_n)=2*a(n), a(c_n)=2*a(n)+1, where p_n = n-th prime, c_n = n-th composite number.

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Haskell

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Maxima

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Formula

A246348 a(1)=1, a(p_n) = 1 + a(n), a(c_n) = 1 + a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also binary width of terms of A135141.

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A246347 Record values in A135141.

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A135141 a(1)=1, a(p_n)=2a(n), a(c_n)=2a(n)+1, where p_n = n-th prime, c_n = n-th composite number.