A266401 - OEIS

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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

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OFFSET

1,2

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

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.]

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 *)

PROG

(PARI)

A030102(n) = { my(r=[n%3]); while(0<n\=3, r=concat(n%3, r)); subst(Polrev(r), x, 3); } \\ After M. 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))))

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.

Sequence in context: A235200 A267099 A266416 * A083798 A197377 A211247

Adjacent sequences: A266398 A266399 A266400 * A266402 A266403 A266404

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Jan 02 2016

STATUS

approved