A026238 -id:A026238 - 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)

A066246 a(n) = 0 unless n is a composite number A002808(k) then a(n) = k.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 4, 5, 0, 6, 0, 7, 8, 9, 0, 10, 0, 11, 12, 13, 0, 14, 15, 16, 17, 18, 0, 19, 0, 20, 21, 22, 23, 24, 0, 25, 26, 27, 0, 28, 0, 29, 30, 31, 0, 32, 33, 34, 35, 36, 0, 37, 38, 39, 40, 41, 0, 42, 0, 43, 44, 45, 46, 47, 0, 48, 49, 50, 0, 51, 0, 52, 53, 54, 55, 56, 0, 57

Offset: 1

Reinhard Zumkeller, Dec 09 2001

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002808, A239968, A010051, A049084.

Cf. A026233, A026238, A065855, A005171.

import Data.List (unfoldr, genericIndex)
a066246 n = genericIndex a066246_list (n - 1)
a066246_list = unfoldr x (1, 1, a002808_list) where
   x (i, z, cs'@(c:cs)) | i == c = Just (z, (i + 1, z + 1, cs))
                        | i /= c = Just (0, (i + 1, z, cs'))
-- Reinhard Zumkeller, Jan 29 2014

Module[{k=1},Table[If[CompositeQ[n],k;k++,0],{n,100}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 03 2019 *)

a(n)=if(isprime(n),0,max(0,n-primepi(n)-1)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = A239968(n) + A010051(n) - 1. - Reinhard Zumkeller, Mar 30 2014

a(n) = A065855(n)*A005171(n). - Ridouane Oudra, Jul 29 2025

A026233 a(n) = j if n is the j-th prime, else a(n) = k if n is the k-th nonprime.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 5, 7, 6, 8, 9, 10, 7, 11, 8, 12, 13, 14, 9, 15, 16, 17, 18, 19, 10, 20, 11, 21, 22, 23, 24, 25, 12, 26, 27, 28, 13, 29, 14, 30, 31, 32, 15, 33, 34, 35, 36, 37, 16, 38, 39, 40, 41, 42, 17, 43, 18, 44, 45, 46, 47, 48

Offset: 1

Clark Kimberling

nonn

Each n occurs at two positions, the distances between them are: 1, 1, 1, 1, 2, 3, 5, 5, 8, 13, 13, 17, 20, 21, 23, 28, 33, 34, 39, 41, 41, ... - Zak Seidov, Mar 06 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A026238.

a026233 n = a049084 n + a239968 n
-- Reinhard Zumkeller, Mar 30 2014, Feb 12 2014

m=100;pr=Prime[Range[m]];npr=Select[Range[m],!PrimeQ[#]&];
a[n_]:=If[PrimeQ[n],PrimePi[n],Position[npr,n][[1,1]]]; Table[a[n],{n,m}] (* Zak Seidov Mar 05 2011 *) s=Range[500];Do[s=Insert[s,n,Prime[n]],{n,100}];s (* Zak Seidov, Mar 05 2011 *)

first(n)=my(p,c); vector(n,k,if(isprime(k),p++,c++)) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A049084(n) + A066246(n) + A000007(A010051(n)). - Reinhard Zumkeller, Feb 12 2014

a(n) = A049084(n) + A239968(n). - Reinhard Zumkeller, Mar 30 2014

A066248 a(n) = if n+1 is prime then A049084(n+1)2 else A066246(n+1)2 - 1.

Original entry on oeis.org

2, 4, 1, 6, 3, 8, 5, 7, 9, 10, 11, 12, 13, 15, 17, 14, 19, 16, 21, 23, 25, 18, 27, 29, 31, 33, 35, 20, 37, 22, 39, 41, 43, 45, 47, 24, 49, 51, 53, 26, 55, 28, 57, 59, 61, 30, 63, 65, 67, 69, 71, 32, 73, 75, 77, 79, 81, 34, 83, 36, 85, 87, 89, 91, 93, 38, 95, 97, 99, 40, 101, 42

Offset: 1

Reinhard Zumkeller, Dec 09 2001

nonn

Permutation of natural numbers; inverse: A066249.

Cf. A066249, A049084, A066246, A026238, A066247, A066250, A066252.

a[n_] := If[PrimeQ[n+1], 2 * PrimePi[n+1], 2 * (n - PrimePi[n+1]) - 1]; Array[a, 100] (* Amiram Eldar, Mar 19 2025 *)

a(n) = A026238(n+1)*2 - A066247(n+1).

A066136 Primes are replaced by their local sequence number in A000040, while composites are replaced by their sequence number in A002808; (a kind of eigen- or home-indexing).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 3, 4, 5, 5, 6, 6, 7, 8, 9, 7, 10, 8, 11, 12, 13, 9, 14, 15, 16, 17, 18, 10, 19, 11, 20, 21, 22, 23, 24, 12, 25, 26, 27, 13, 28, 14, 29, 30, 31, 15, 32, 33, 34, 35, 36, 16, 37, 38, 39, 40, 41, 17, 42, 18, 43, 44, 45, 46, 47, 19, 48, 49, 50, 20, 51, 21, 52, 53

Offset: 1

Labos Elemer, Dec 07 2001

nonn

Primality or compositeness is recognizable from the terms: if successor of a term(>2) is larger by one than the term, then it labels a composite number.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A000040, A000720, A002808, A026238.

Do[s=n; If[PrimeQ[n], Print[PrimePi[n]]]; If[ !PrimeQ[n], Print[n-PrimePi[n]-1]], {n, 1, 1000}]

a(n) = { if (isprime(n), primepi(n), n - primepi(n) - 1) } \\ Harry J. Smith, Feb 02 2010

a(n) = pi(n) if n is prime; a(n) = n-pi(n)-1 if n is not prime.

a(n) = A026238(n), n>1. - R. J. Mathar, Sep 30 2008

A066250 a(n) = if n+1 is prime then A049084(n+1)2 - 1 else A066246(n+1)2.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 8, 10, 9, 12, 11, 14, 16, 18, 13, 20, 15, 22, 24, 26, 17, 28, 30, 32, 34, 36, 19, 38, 21, 40, 42, 44, 46, 48, 23, 50, 52, 54, 25, 56, 27, 58, 60, 62, 29, 64, 66, 68, 70, 72, 31, 74, 76, 78, 80, 82, 33, 84, 35, 86, 88, 90, 92, 94, 37, 96, 98, 100, 39, 102

Offset: 1

Reinhard Zumkeller, Dec 09 2001

nonn

Permutation of natural numbers; inverse: A066251.

Index entries for sequences that are permutations of the natural numbers

Cf. A066251, A049084, A066246, A026238, A010051, A066248, A066254.

a(n) = A026238(n+1)*2 - A010051(n+1).

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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A066246 a(n) = 0 unless n is a composite number A002808(k) then a(n) = k.

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A026233 a(n) = j if n is the j-th prime, else a(n) = k if n is the k-th nonprime.

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A066248 a(n) = if n+1 is prime then A049084(n+1)*2 else A066246(n+1)*2 - 1.

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A066136 Primes are replaced by their local sequence number in A000040, while composites are replaced by their sequence number in A002808; (a kind of eigen- or home-indexing).

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A066250 a(n) = if n+1 is prime then A049084(n+1)*2 - 1 else A066246(n+1)*2.

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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.

A066248 a(n) = if n+1 is prime then A049084(n+1)2 else A066246(n+1)2 - 1.

A066250 a(n) = if n+1 is prime then A049084(n+1)2 - 1 else A066246(n+1)2.