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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] 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 November 30 21:57 EST 2022. Contains 358453 sequences. (Running on oeis4.)