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A135044 a(1)=1, then a(c) = p and a(p) = c, where c = T_c(r,k) and p = T_p(r,k), and where T_p contains the primes arranged in rows by the prime index chain and T_c contains the composites arranged in rows by the order of compositeness. See Formula. 5
1, 4, 9, 2, 16, 7, 6, 13, 3, 19, 26, 17, 8, 23, 41, 5, 12, 67, 10, 29, 59, 37, 14, 83, 179, 11, 43, 331, 20, 47, 39, 109, 277, 157, 53, 431, 22, 1063, 31, 191, 15, 2221, 27, 61, 211, 71, 30, 599, 1787, 919, 241, 3001, 35, 73, 8527, 127, 1153, 79, 21, 19577, 44, 89, 283 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Exchanges primes with composites, primeth primes with composith composites, etc.

Exchange the k-th prime of order j with the k-th composite of order j and vice versa.

Self-inverse permutation of positive integers.

If n is the composite number A236536(r,k), then a(n) is the corresponding prime A236542(r,k) at the same position (r,k). Vice versa, if n is the prime A236542(r,k), then a(n) is the corresponding composite A236536(r,k) at the same position. - Andrew Weimholt, Jan 28 2014

The original name for this entry did not produce this sequence, but instead A236854, which differs from this permutation for the first time at n=8, where A236854(8)=23, while here a(8)=13. - Antti Karttunen, Feb 01 2014

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..197

N. Fernandez, An order of primeness, F(p).

N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Index to permutations of positive integers

FORMULA

a(1)=1, a(A236536(r,k))=A236542(r,k), a(A236542(r,k))=A236536(r,k)

EXAMPLE

From Andrew Weimholt, Jan 29 2014: (Start)

More generally, takes the primes organized in an array according to the sieving process described in the Fernandez paper:

        Row[1](n) = 2, 7, 13, 19, 23, ...

        Row[2](n) = 3, 17, 41, 67, 83, ...

        Row[3](n) = 5, 59, 179, ...

        Row[4](n) = 11, 277, ...

        Lets call this  T_p (n, k)

Also take the composites organized in a similar manner, except we use "composite" numbered positions in our sieve:

        Row[1](n) = 4, 6, 8, 10, 14, 20, 22, ...

        Row[2](n) = 9, 12, 15, 18, 24, ...

        Row[3](n) = 16, 21, 25, ...

        Lets call this T_c (n, k)

If we now take the natural numbers and swap each number (except for 1) with the number which holds the same spot in the other array, then we get the sequence: 1, 4, 9, 2, 16, 7, 6, 13, with for example a(8) = 13 (13 holds the same position in the 'prime' table as 8 does in the 'composite' table). (End)

MAPLE

A135044 := proc(n)

    if n = 1 then

        1;

    elif isprime(n) then

        idx := -1 ;

        for r from 1 do

            for c from 1 do

                if A236542(r, c) = n then

                    idx := [r, c] ;

                end if;

                if A236542(r, c) >= n then

                    break;

                end if;

            end do:

            if type(idx, list)  then

                break;

            end if;

        end do:

        A236536(r, c) ;

    else

        idx := -1 ;

        for r from 1 do

            for c from 1 do

                if A236536(r, c) = n then

                    idx := [r, c] ;

                end if;

                if A236536(r, c) >= n then

                    break;

                end if;

            end do:

            if type(idx, list)  then

                break;

            end if;

        end do:

        A236542(r, c) ;

    end if;

end proc: # R. J. Mathar, Jan 28 2014

MATHEMATICA

Composite[n_Integer] := Block[{k = n + PrimePi@n + 1}, While[k != n + PrimePi@k + 1, k++ ]; k]; Compositeness[n_] := Block[{c = 1, k = n}, While[ !(PrimeQ@k || k == 1), k = k - 1 - PrimePi@k; c++ ]; c]; Primeness[n_] := Block[{c = 1, k = n}, While[ PrimeQ@k, k = PrimePi@k; c++ ]; c];

ckj[k_, j_] := Select[ Table[Composite@n, {n, 10000}], Compositeness@# == j &][[k]]; pkj[k_, j_] := Select[ Table[Prime@n, {n, 3000}], Primeness@# == j &][[k]]; f[0]=0; f[1] = 1;

f[n_] := If[ PrimeQ@ n, pn = Primeness@n; ckj[ Position[ Select[ Table[ Prime@ i, {i, 150}], Primeness@ # == pn &], n][[1, 1]], pn], cn = Compositeness@n; pkj[ Position[ Select[ Table[ Composite@ i, {i, 500}], Compositeness@ # == cn &], n][[1, 1]], cn]]; Array[f, 64] (* Robert G. Wilson v *)

CROSSREFS

Cf. A000040, A007097, A049076, A049078 - A049081, A058322, A058324 - A058328, A093046, A002808, A006508, A059981, A078442, A236854.

Sequence in context: A128204 A079049 A114578 * A236854 A235491 A256513

Adjacent sequences:  A135041 A135042 A135043 * A135045 A135046 A135047

KEYWORD

nonn

AUTHOR

Katarzyna Matylla, Feb 11 2008

EXTENSIONS

Edited, corrected and extended by Robert G. Wilson v, Feb 18 2008

Name corrected by Andrew Weimholt, Jan 29 2014

STATUS

approved

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)