%I #44 Mar 17 2021 04:19:00
%S 1,4,9,2,16,7,6,23,3,53,26,17,14,13,83,5,12,241,35,101,59,43,8,41,431,
%T 11,37,1523,75,149,39,547,277,191,19,179,27,3001,31,157,24,12763,22,
%U 379,859,167,114,3943,1787,1153,67,1063,10,103,27457,127,919,89,21
%N Self-inverse permutation of natural numbers: a(1)=1, then a(p_n)=c_{a(n)}, a(c_n)=p_{a(n)}, where p_n = n-th prime, c_n = n-th composite.
%C Shares with A026239 the property that the prime-positions 2, 3, 5, 7, ... contain only composite values and the composite-positions 4, 6, 8, 9, ..., etc. contain only prime values. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, so this is a kind of "deep" variant of A026239. Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are entangled with each other. In this case a complementary pair primes/composites (A000040/A002808) is entangled with a complementary pair composites/primes.
%C Maps A006508 to A007097 and vice versa.
%H Chai Wah Wu, <a href="/A236854/b236854.txt">Table of n, a(n) for n = 1..735</a> (n = 1..150 from Alois P. Heinz)
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(1)=1, a(p_i) = A002808(a(i)) for primes with index i, a(c_j) = A000040(a(j)) for composites with index j (where 4 has index 1, 6 has index 2, and so on).
%e a(5)=c(a(3))=c(9)=16, because 5=prime(3), and the 9th composite number is c(9)=16.
%e Thus a(10)=prime(a(5))=prime(16)=53 (since 10 is the 5th composite), a(18)=prime(a(10))=prime(53)=241 (since 18 is the 10th composite), a(28)=prime(a(18))=prime(241)=1523.
%e A significant record value is a(198) = prime(a(152)) = prime(563167303) since 198=c(152); a(152)=prime(a(115)) since 152=c(115); a(115)=prime(a(84)); a(84)=prime(a(60)); a(60)=prime(a(42)); a(42)=prime(a(28)).
%t terms = 150; cc = Select[Range[4, 2 terms^2(*empirical*)], CompositeQ]; compositePi[k_?CompositeQ] := FirstPosition[cc, k][[1]]; a[1] = 1; a[p_?PrimeQ] := a[p] = cc[[a[PrimePi[p]]]]; a[k_] := a[k] = Prime[a[ compositePi[k]]]; Array[a, terms] (* _Jean-François Alcover_, Mar 02 2016 *)
%o (Scheme, with memoization-macro definec from _Antti Karttunen_'s IntSeq-library)
%o (definec (A236854 n) (cond ((< n 2) n) ((prime? n) (A002808 (A236854 (A000720 n)))) (else (A000040 (A236854 (A065855 n))))))
%o (PARI) A236854(n)={if(isprime(n), A002808(A236854(primepi(n))), n==1&&return(1);prime(A236854(n-primepi(n)-1)))} \\ without memoization: not much slower. - _M. F. Hasler_, Feb 03 2014
%o (PARI) a236854=vector(999);a236854[1]=1;A236854(n)={a236854[n]&&return(a236854[n]); a236854[n]=if(isprime(n), A002808(A236854(primepi(n))), prime(A236854(n-primepi(n)-1)))} \\ Version with memoization. - _M. F. Hasler_, Feb 03 2014
%o (Python)
%o from sympy import primepi, prime, isprime
%o def a002808(n):
%o m, k = n, primepi(n) + 1 + n
%o while m != k: m, k = k, primepi(k) + 1 + n
%o return m # this function from _Chai Wah Wu_
%o def a(n): return n if n<2 else a002808(a(primepi(n))) if isprime(n) else prime(a(n - primepi(n) - 1))
%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 07 2017
%Y Differs from A135044 for the first time at n=8, where A135044(8)=13, while here a(8)=23.
%Y Cf. A026239, A135141/A227413, A006508, A007097, A000040, A002808, A000720, A065855.
%K nonn
%O 1,2
%A _Antti Karttunen_, Feb 01 2014, based on _Katarzyna Matylla_'s original but misplaced definition for A135044 from Feb 11 2008.
%E Values double-checked by _M. F. Hasler_, Feb 03 2014