1,2

This is a multiplicative self-inverse permutation of the integers.

A225547gives the fixed points.

FromAntti KarttunenandPeter Munn,Feb 02 2020: (Start)

This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, inA329050,this sequence reflects these factors about the main diagonal of the array, substitutingA329050[j,i] forA329050[i,j], and this results in many relationships including significant homomorphisms.

This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of theA329050array is the square of the previous column, and each successive row is the prime shift of the previous row.

A329050gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.

Alternative construction: For n > 1, form a vector v of lengthA299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)}A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponentA048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary stringA087207(n).

This permutation effects the following mappings:

A000035(a(n)) =A010052(n),A010052(a(n)) =A000035(n). [Odd numbers <-> Squares]

A008966(a(n)) =A209229(n),A209229(a(n)) =A008966(n). [Squarefree numbers <-> Powers of 2]

(End)

FromAntti Karttunen,Jul 08 2020: (Start)

Moreover, we see also that this sequence maps betweenA016825(Numbers of the form 4k+2) andA001105(2*squares) as well as betweenA008586(Multiples of 4) andA028983(Numbers with even sum of the divisors).

(End)

Paul Tek,Table of n, a(n) for n = 1..40

Multiplicative, with a(prime(i)^j) =A019565(j)^A000079(i-1).

a(prime(i)) = 2^(2^(i-1)).

FromAntti KarttunenandPeter Munn,Feb 06 2020: (Start)

a(A329050(n,k)) =A329050(k,n).

a(A329332(n,k)) =A329332(k,n).

Equivalently, a(A019565(n)^k) =A019565(k)^n. If n = 1, this gives a(2^k) =A019565(k).

a(A059897(n,k)) =A059897(a(n), a(k)).

The previous formula implies a(n*k) = a(n) * a(k) ifA059895(n,k) = 1.

a(A000040(n)) =A001146(n-1); a(A001146(n)) =A000040(n+1).

a(A000290(a(n))) =A003961(n); a(A003961(a(n))) =A000290(n) = n^2.

a(A000265(a(n))) =A008833(n); a(A008833(a(n))) =A000265(n).

a(A006519(a(n))) =A007913(n); a(A007913(a(n))) =A006519(n).

A007814(a(n)) =A248663(n);A248663(a(n)) =A007814(n).

A048675(a(n)) =A048675(n) andA048675(a(2^k * n)) =A048675(2^k * a(n)) = k +A048675(a(n)).

(End)

FromAntti KarttunenandPeter Munn,Jul 08 2020: (Start)

For all n >= 1, a(2n) =A334747(a(n)).

In particular, for n =A003159(m), m >= 1, a(2n) = 2*a(n). [Note thatA003159includes all odd numbers]

(End)

7744 = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).

a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (*Michael De Vlieger,Jan 21 2020 *)

(PARI)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));

a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i, 1]); f[i, 1] =A019565(f[i, 2]); f[i, 2] = 2^(primepi(p)-1); ); factorback(f); } \\Michel Marcus,Nov 29 2019

(PARI)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

A225546(n) = if(1==n, 1, my(f=factor(n), u=#binary(vecmax(f[, 2])), prods=vector(u, x, 1), m=1, e); for(i=1, u, for(k=1, #f~, if(bitand(f[k, 2], m), prods[i] *= f[k, 1])); m<<=1); prod(i=1, u, prime(i)^A048675(prods[i]))); \\Antti Karttunen,Feb 02 2020

(Python)

from math import prod

from sympy import prime, primepi, factorint

defA225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1], 1) if v == '1')**(1<<primepi(p)-1) for p, e in factorint(n).items()) #Chai Wah Wu,Mar 17 2023

Cf.A225547(fixed points) and the subsequences listed there.

TransposesA329050,A329332.

An automorphism of positive integers under the binary operationsA059895,A059896,A059897,A306697,A329329.

An automorphism ofA059897subgroups:A000379,A003159,A016754,A122132.

Permutes lists where membership is determined by number of Fermi-Dirac factors:A000028,A050376,A176525,A268388.

Also permutes:A036554,A059404,A066427,A066428,A072774,A331593.

Sequences f that satisfy f(a(n)) = f(n):A048675,A064179,A064547,A097248,A302777,A331592.

Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).

Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).

Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: (A000035,A010052), (A008966,A209229), (A007814,A248663), (A061395,A299090), (A087207,A267116), (A225569,A227291).

Cf.A331287[= gcd(a(n),n)].

Cf.A331288[= min(a(n),n)], see alsoA331301.

Cf.A331309[=A000005(a(n)), number of divisors].

Cf.A331590[= a(a(n)*a(n))].

Cf.A331591[=A001221(a(n)), number of distinct prime factors], see alsoA331593.

Cf.A331740[=A001222(a(n)), number of prime factors with multiplicity].

Cf.A331733[=A000203(a(n)), sum of divisors].

Cf.A331734[=A033879(a(n)), deficiency].

Cf.A331735[=A009194(a(n))].

Cf.A331736[=A000265(a(n)) = a(A008833(n)), largest odd divisor].

Cf.A335914[=A038040(a(n))].

A self-inverse isomorphism between pairs ofA059897subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).

A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825,A001105), (A008586,A028983).

Cf. alsoA336321,A336322(compositions with another involution,A122111).

Sequence in context:A334866A308317A318363*A329050A053124A242500

Adjacent sequences:A225543A225544A225545*A225547A225548A225549

nonn,mult

Paul Tek,May 10 2013

Name edited byPeter Munn,Feb 14 2020

"Tek's flip" prepended to the name byAntti Karttunen,Jul 08 2020

approved