A026233 -id:A026233 - cp's OEIS frontend

A269857 Permutation of natural numbers: a(1) = 1; if n is an odd prime, a(n) = A250469(a(A026233(n))), else a(n) = 2*(a(A026233(n))).

1, 2, 3, 4, 5, 6, 9, 8, 10, 12, 7, 18, 15, 16, 20, 24, 25, 14, 21, 36, 30, 32, 27, 40, 48, 50, 28, 42, 33, 72, 11, 60, 64, 54, 80, 96, 51, 100, 56, 84, 35, 66, 45, 144, 22, 120, 57, 128, 108, 160, 192, 102, 69, 200, 112, 168, 70, 132, 49, 90, 39, 288, 44, 240, 114, 256, 55, 216, 320, 384, 105, 204, 87, 138, 400

Offset: 1

Antti Karttunen, Mar 06 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2662
Index entries for sequences that are permutations of the natural numbers

Inverse: A269858.
Cf. A005843, A026233, A010051, A250469.
Related or similar permutations: A071574, A252755, A269847.
Differs from A269847 for the first time at n=19, where a(19)=21, while A269847(19)=27.

a(1) = 1; if n is an odd prime, a(n) = A250469(a(A026233(n))), else a(n) = 2*(a(A026233(n))).
As a composition of other permutations:
a(n) = A252755(A071574(n)).

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

A239968 a(n) = 0 unless n is a nonprime A018252(k) then a(n) = k.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 30 2014

nonn

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

Cf. A005171, A010051, A018252, A057427, A066246, A049084, A026233, A062298.

import Data.List (unfoldr, genericIndex)
a239968 n = genericIndex a239968_list (n - 1)
a239968_list = unfoldr c (1, 1, a018252_list) where
   c (i, z, xs'@(x:xs)) | i == x = Just (z, (i + 1, z + 1, xs))
                        | i /= x = Just (0, (i + 1, z, xs'))

Module[{k = 0}, Array[If[!PrimeQ[#], ++k, 0] &, 100]] (* Paolo Xausa, Jul 31 2025 *)

a(n) = A066246(n) - A010051(n) + 1.
a(n) = A026233(n) - A049084(n);
A057427(a(n)) = A005171(n).
a(n) = A062298(n)*A005171(n). - Ridouane Oudra, Jul 29 2025

A026238 a(n) = j if n is the j-th prime, else a(n) = k if n is the k-th composite.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A049084, A066246, A026233, A239968.

a026238 n = a049084 n + a066246 n  -- Reinhard Zumkeller, Jan 29 2014

Table[If[PrimeQ[n],PrimePi[n],n-1-PrimePi[n]],{n,2,70}] (* Harvey P. Dale, Jun 06 2017 *)

first(n)=my(c,p); vector(n-1,k, if(isprime(k+1),p++,c++)) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A049084(n) + A066246(n) for n >= 2.

A014237 a(n) = (n-th prime) - (n-th nonprime).

Original entry on oeis.org

1, -1, -1, -1, 2, 3, 5, 5, 8, 13, 13, 17, 20, 21, 23, 28, 33, 34, 39, 41, 41, 46, 49, 54, 61, 63, 64, 67, 67, 69, 82, 85, 89, 90, 99, 100, 105, 109, 112, 117, 122, 123, 131, 131, 134, 135, 146, 157, 159, 160, 163, 167, 167, 176, 181, 186, 191, 191, 196

Offset: 1

Clark Kimberling

sign

a(n) = A000040(n) - A018252(n). - Reinhard Zumkeller, Apr 30 2014

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

Cf. A038529, A026233, A127118.

a014237 n = a000040 n - a018252 n  -- Reinhard Zumkeller, Apr 30 2014

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n+PrimePi@n];
Table[Prime[n] - nonPrime[n], {n, 1, 70}] (* G. C. Greubel, Jun 22 2019 *)

from sympy import prime, composite
def A014237(n):
    return 1 if n == 1 else prime(n)-composite(n-1) # Chai Wah Wu, Dec 27 2018

A269847 Permutation of natural numbers: a(1) = 1, for n > 1, if n is an odd prime, a(n) = A003961(a(A000720(n))), otherwise a(n) = 2*a(n-A000720(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 10, 12, 7, 18, 15, 16, 20, 24, 25, 14, 27, 36, 30, 32, 21, 40, 48, 50, 28, 54, 45, 72, 11, 60, 64, 42, 80, 96, 75, 100, 56, 108, 35, 90, 81, 144, 22, 120, 63, 128, 84, 160, 192, 150, 135, 200, 112, 216, 70, 180, 49, 162, 33, 288, 44, 240, 126, 256, 125, 168, 320, 384, 225, 300, 105, 270, 400

Offset: 1

Antti Karttunen, Mar 06 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A269848.
Cf. A000040, A000720, A003961, A005843, A007097, A010051, A026233, A065090, A065091.
Related or similar permutations: A071574, A163511, A246681, A257730, A269857.

a(1) = 1, and for n > 1, if n is an odd prime, a(n) = A003961(a(A000720(n))), otherwise [when n is 2 or composite] a(n) = 2*a(n-A000720(n)).
a(1) = 1; if n is an odd prime, a(n) = A003961(a(A026233(n))), else a(n) = A005843(a(A026233(n))).
Declarative definition:
a(1)=1, a(A065091(n)) = A003961(a(n+1)), a(A065090(n+1)) = 2*a(n).
As a composition of other permutations:
a(n) = A163511(A071574(n)).
Other identities. For all n >= 1:
a(A007097(n)) = A000040(n). [Maps the terms of primeth recurrence to primes.]

A244724 Lexicographically earliest permutation of the natural numbers such that primes and composites alternate in the sums of adjacent terms.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 8, 9, 7, 10, 11, 12, 13, 16, 14, 15, 17, 20, 18, 19, 21, 22, 23, 24, 25, 28, 26, 27, 29, 30, 32, 35, 31, 36, 33, 34, 38, 41, 37, 42, 39, 40, 44, 45, 43, 46, 47, 50, 48, 49, 51, 52, 53, 54, 56, 57, 55, 58, 59, 68, 60, 67, 61, 66, 62, 65, 63

Offset: 1

Reinhard Zumkeller, Jul 05 2014

nonn

For k > 0: a(2*k-1) + a(2*k) is prime, a(2*k) + a(2*k+1) is composite.

			.             n | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
.          a(n) | 1 2 4 3 5 6 8 9 7 10 11 12 13 16 14 15 17 20 18 19
. A026233(a(n)) | 1 1 2 2 3 3 4 5 4  6  5  7  6 10  8  9  7 12 11  8 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A244732 (inverse), A244731 (fixed points), A073846, A113321, A115316.

import Data.List (delete)
a244724 n = a244724_list !! (n-1)
a244724_list = 1 : f 1 [2..] where
   f x xs = f' xs where
     f' (u:us) | a010051' (x + u) == 1 = g u (delete u xs)
               | otherwise             = f' us where
        g y ys = g' ys where
          g' (v:vs) | a010051' (y + v) == 0 = u : v : f v (delete v ys)
                    | otherwise        = g' vs

A010051(a(n)+a(n+1)) = n mod 2.

A249594 Positions of primes in A249054.

Original entry on oeis.org

2, 4, 7, 11, 12, 14, 17, 18, 19, 22, 23, 27, 30, 32, 34, 37, 38, 41, 42, 45, 47, 50, 51, 53, 54, 55, 58, 61, 64, 65, 68, 71, 72, 74, 75, 78, 80, 81, 84, 87, 89, 90, 94, 97, 100, 102, 105, 108, 109, 111, 113, 116, 117, 119, 120, 123, 125, 129, 133, 134, 135

Offset: 1

Reinhard Zumkeller, Nov 03 2014

nonn

A249054(a(n)) = A000040(n); sequence is strictly increasing, i.e. all primes occur in A249054 in natural order, see also A249595.

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

Cf. A000040, A026233, A049084, A239968, A249054, A249595 (complement).

import Data.List (elemIndex); import Data.Maybe (fromJust)
a249594 = (+ 1) . fromJust . (`elemIndex` a249054_list) . a000040

A249595 Positions of nonprimes in A249054.

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 10, 13, 15, 16, 20, 21, 24, 25, 26, 28, 29, 31, 33, 35, 36, 39, 40, 43, 44, 46, 48, 49, 52, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 73, 76, 77, 79, 82, 83, 85, 86, 88, 91, 92, 93, 95, 96, 98, 99, 101, 103, 104, 106, 107, 110, 112, 114, 115

Offset: 1

Reinhard Zumkeller, Nov 03 2014

nonn

A249054(a(n)) = A018252(n); sequence is strictly increasing, i.e. all nonprimes occur in A249054 in natural order, see also A249594.

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

Cf. A018252, A026233, A049084, A239968, A249054, A249594 (complement).

import Data.List (elemIndex); import Data.Maybe (fromJust)
a249595 = (+ 1) . fromJust . (`elemIndex` a249054_list) . a018252

A102885 Index of n in the primes A000040 or nonprimes A141468.

Original entry on oeis.org

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

Offset: 0

Juri-Stepan Gerasimov, Aug 17 2008

The nonnegative numbers n occur exactly once in either A000040 or A141468. The sequence lists the corresponding index. It is a permutation of A008619.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A141468, A000040, A026233.

Module[{nn=80,pr,np},pr=Prime[Range[PrimePi[nn]]];np=Complement[ Range[ 0,nn],pr];Table[If[PrimeQ[n],Position[pr,n],Position[np,n]],{n,0,nn}]]//Flatten (* Harvey P. Dale, Sep 10 2022 *)

A141468(a(n))=n or A000040(a(n))=n.

Edited by R. J. Mathar, Aug 19 2008

cp's OEIS Frontend

A269857 Permutation of natural numbers: a(1) = 1; if n is an odd prime, a(n) = A250469(a(A026233(n))), else a(n) = 2*(a(A026233(n))).

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

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A239968 a(n) = 0 unless n is a nonprime A018252(k) then a(n) = k.

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A026238 a(n) = j if n is the j-th prime, else a(n) = k if n is the k-th composite.

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A014237 a(n) = (n-th prime) - (n-th nonprime).

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A269847 Permutation of natural numbers: a(1) = 1, for n > 1, if n is an odd prime, a(n) = A003961(a(A000720(n))), otherwise a(n) = 2*a(n-A000720(n)).

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A244724 Lexicographically earliest permutation of the natural numbers such that primes and composites alternate in the sums of adjacent terms.

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A249594 Positions of primes in A249054.

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A249595 Positions of nonprimes in A249054.

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