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%I #101 Mar 17 2023 13:19:22
%S 1,2,4,3,16,8,256,6,9,32,65536,12,4294967296,512,64,5,
%T 18446744073709551616,18,340282366920938463463374607431768211456,48,
%U 1024,131072,115792089237316195423570985008687907853269984665640564039457584007913129639936,24,81,8589934592,36,768
%N Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).
%C This is a multiplicative self-inverse permutation of the integers.
%C A225547 gives the fixed points.
%C From _Antti Karttunen_ and _Peter Munn_, Feb 02 2020: (Start)
%C This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.
%C This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.
%C A329050 gives 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.
%C Alternative construction: For n > 1, form a vector v of length A299090(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 exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).
%C This permutation effects the following mappings:
%C A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]
%C A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]
%C (End)
%C From _Antti Karttunen_, Jul 08 2020: (Start)
%C Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).
%C (End)
%H Paul Tek, <a href="/A225546/b225546.txt">Table of n, a(n) for n = 1..40</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F Multiplicative, with a(prime(i)^j) = A019565(j)^A000079(i-1).
%F a(prime(i)) = 2^(2^(i-1)).
%F From _Antti Karttunen_ and _Peter Munn_, Feb 06 2020: (Start)
%F a(A329050(n,k)) = A329050(k,n).
%F a(A329332(n,k)) = A329332(k,n).
%F Equivalently, a(A019565(n)^k) = A019565(k)^n. If n = 1, this gives a(2^k) = A019565(k).
%F a(A059897(n,k)) = A059897(a(n), a(k)).
%F The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1.
%F a(A000040(n)) = A001146(n-1); a(A001146(n)) = A000040(n+1).
%F a(A000290(a(n))) = A003961(n); a(A003961(a(n))) = A000290(n) = n^2.
%F a(A000265(a(n))) = A008833(n); a(A008833(a(n))) = A000265(n).
%F a(A006519(a(n))) = A007913(n); a(A007913(a(n))) = A006519(n).
%F A007814(a(n)) = A248663(n); A248663(a(n)) = A007814(n).
%F A048675(a(n)) = A048675(n) and A048675(a(2^k * n)) = A048675(2^k * a(n)) = k + A048675(a(n)).
%F (End)
%F From _Antti Karttunen_ and _Peter Munn_, Jul 08 2020: (Start)
%F For all n >= 1, a(2n) = A334747(a(n)).
%F In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers]
%F (End)
%e 7744 = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
%e a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.
%t 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 *)
%o (PARI)
%o A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
%o 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
%o (PARI)
%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o 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
%o (Python)
%o from math import prod
%o from sympy import prime, primepi, factorint
%o def A225546(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
%Y Cf. A225547 (fixed points) and the subsequences listed there.
%Y Transposes A329050, A329332.
%Y An automorphism of positive integers under the binary operations A059895, A059896, A059897, A306697, A329329.
%Y An automorphism of A059897 subgroups: A000379, A003159, A016754, A122132.
%Y Permutes lists where membership is determined by number of Fermi-Dirac factors: A000028, A050376, A176525, A268388.
%Y Also permutes: A036554, A059404, A066427, A066428, A072774, A331593.
%Y Sequences f that satisfy f(a(n)) = f(n): A048675, A064179, A064547, A097248, A302777, A331592.
%Y Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).
%Y Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).
%Y 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).
%Y Cf. A331287 [= gcd(a(n),n)].
%Y Cf. A331288 [= min(a(n),n)], see also A331301.
%Y Cf. A331309 [= A000005(a(n)), number of divisors].
%Y Cf. A331590 [= a(a(n)*a(n))].
%Y Cf. A331591 [= A001221(a(n)), number of distinct prime factors], see also A331593.
%Y Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity].
%Y Cf. A331733 [= A000203(a(n)), sum of divisors].
%Y Cf. A331734 [= A033879(a(n)), deficiency].
%Y Cf. A331735 [= A009194(a(n))].
%Y Cf. A331736 [= A000265(a(n)) = a(A008833(n)), largest odd divisor].
%Y Cf. A335914 [= A038040(a(n))].
%Y A self-inverse isomorphism between pairs of A059897 subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849 U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850 U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).
%Y A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825, A001105), (A008586, A028983).
%Y Cf. also A336321, A336322 (compositions with another involution, A122111).
%K nonn,mult
%O 1,2
%A _Paul Tek_, May 10 2013
%E Name edited by _Peter Munn_, Feb 14 2020
%E "Tek's flip" prepended to the name by _Antti Karttunen_, Jul 08 2020
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