A266401 Self-inverse permutation of natural numbers: a(n) = A064989(A263273(A003961(n))).

1, 2, 5, 4, 3, 10, 17, 8, 13, 6, 11, 20, 9, 34, 71, 16, 7, 26, 19, 12, 23, 22, 21, 40, 41, 18, 227, 68, 31, 142, 29, 32, 53, 14, 67, 52, 61, 38, 107, 24, 25, 46, 59, 44, 65, 42, 73, 80, 49, 82, 197, 36, 33, 454, 55, 136, 137, 62, 43, 284, 37, 58, 571, 64, 45, 106, 35, 28, 89, 134, 15, 104, 47

Offset: 1

Author

Antti Karttunen, Jan 02 2016

Keywords

• nonn
• base

Comments

Shift primes in the prime factorization of n one step towards larger primes (A003961), then apply the bijective base-3 reverse (A263273) to the resulting odd number, which yields another (or same) odd number, then shift primes in the prime factorization of that second odd number one step back towards smaller primes (A064989).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..6560
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000035, A003961, A064989, A263273.

Cf. A265369, A265904, A266190, A266403 (other conjugates or similar sequences derived from A263273).

Cf. also A048673, A064216, A264996, A266402, A266407, A266408.

Programs

Mathematica

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; g[p_?PrimeQ] := g[p] = Prime[PrimePi@ p + 1]; g[1] = 1; g[n_] := g[n] = Times @@ (g[First@ #]^Last@ # &) /@ FactorInteger@ n; h[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Table[h@ f@ g@ n, {n, 82}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 and A263273 *)

PARI

A030102(n) = { my(r=[n%3]); while(0M. F. Hasler's Nov 04 2011 code in A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3))); \\ Taking of the quotient probably unnecessary.
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A266401 = n -> A064989(A263273(A003961(n)));
for(n=1, 6560, write("b266401.txt", n, " ", A266401(n)));

Scheme

(define (A266401 n) (A064989 (A263273 (A003961 n))))

Formula

a(n) = A064989(A263273(A003961(n))).

As a composition of related permutations:

a(n) = A064216(A264996(A048673(n))).

Other identities. For all n >= 0:

A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]