A293448 - OEIS

%I #26 Nov 24 2017 04:32:59

%S 1,2,3,4,5,6,7,8,9,10,11,18,13,14,15,16,17,12,19,50,21,22,23,54,25,26,

%T 27,98,29,30,31,32,33,34,35,36,37,38,39,250,41,70,43,242,75,46,47,162,

%U 49,20,51,338,53,24,55,686,57,58,59,150,61,62,147,64,65,154,67,578,69,42,71,108,73,74,45,722,77,286,79,1250,81,82,83

%N Self-inverse permutation of natural numbers: replace (with multiplicity) each prime factor A000040(k) with A000040(min+(max-k)) in the prime factorization of n, where min = A055396(n) and max = A061395(n).

%C Reverse the prime-indices in such a way that the smallest and the greatest prime dividing n (A020639 and A006530) are preserved.

%C a(n) = n iff n belongs to A236510. - _Rémy Sigrist_, Nov 22 2017

%H Antti Karttunen, <a href="/A293448/b293448.txt">Table of n, a(n) for n = 1..8192</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F For all even squarefree numbers coincides with A273258, that is, for all n, a(A039956(n)) = A273258(A039956(n)).

%e For n = 25 = 5^2 = prime(3)^2, thus min = max = 3, and we form a product prime(3+(3-3))^2, thus a(25) = prime(3)^2 = 25.

%e For n = 42 = 2*3*7 = prime(1)*prime(2)*prime(4), thus min = 1 and max = 4, so we form a product prime(1+(4-1))*prime(1+(4-2))*prime(1+(4-4)), thus a(42) = prime(4)*prime(3)*prime(1) = 7*5*2 = 70.

%e For n = 126 = 2 * 3^2 * 7 = prime(1) * prime(2)^2 * prime(4), thus min = 1 and max = 4, so we form a product prime(1+(4-1)) * prime(1+(4-2))^2 * prime(1+(4-4)), thus a(126) = prime(4) * prime(3)^2 * prime(1) = 7 * 5^2 * 2 = 350.

%o (PARI) A293448(n) = { if(1==n,return(n)); my(f=factor(n), mini = primepi(f[1, 1]), maxi = primepi(f[#f~, 1])); for(i=1,#f~,f[i,1] = prime((maxi-primepi(f[i,1]))+mini)); factorback(f); }

%Y Cf. A000720, A055396, A057889, A061395, A236510 (fixed points), A273258.

%Y Differs from A069799 (and some other related permutations) for the first time at n=42.

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 09 2017