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%I #26 Dec 11 2022 10:31:17
%S 1,4,9,2,25,12,49,16,3,20,121,6,169,28,45,8,289,18,361,10,63,44,529,
%T 36,5,52,81,14,841,60,961,64,99,68,175,24,1369,76,117,50,1681,84,1849,
%U 22,15,92,2209,48,7,40,153,26,2809,72,275,98,171,116,3481,30,3721,124,21,32
%N Each term a(n) satisfies four properties: 1, divisible by all prime factors of n; 2, divisible by only the prime factors of n; 3, not equal to any of the terms a(1), a(2), ... a(n-1); 4, smallest number satisfying 1-3 if A005361(n) is even, or second smallest number satisfying 1-3 if A005361(n) is odd.
%C This sequence is permutation of the positive integers.
%C The prime p occurs at n = p^2.
%C Multiples of a number x have density 1/x.
%C Conjecture: this permutation of positive integers is self-inverse. Compare with A358971. The principal distinction between this sequence and A358971 is that fixed points aside from A358971(1) = 1 are explicitly ruled out in the latter. - _Michael De Vlieger_, Dec 10 2022
%D Brad Klee, Posting to Sequence Fans Mailing List, Dec 21, 2014.
%H Michael De Vlieger, <a href="/A253288/b253288.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A253288/a253288.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H Michael De Vlieger, <a href="/A253288/a253288_1.png">Log log scatterplot of a(n) <= 12000</a>, n = 1..2^10 showing primes in red, other prime powers (in A246547) in gold, squarefree composites (in A120944) in green, numbers neither squarefree nor prime power (in A120706) in blue and magenta. The terms in magenta are products of composite prime powers (in A286708).
%H Michael De Vlieger, <a href="/A253288/a253288_2.png">Log log scatterplot of a(n) <= 2^14</a>, n = 1..2^14, showing a(n) such that rad(n) = 6 in red, and A358971(n) such that rad(n) = 6 in blue for comparison. This is an example of a self-inverse relation among terms a(n) in A003586.
%H Michael De Vlieger, <a href="/A253288/a253288_3.png">Log log scatterplot of a(n) <= 80000</a>, n = 1..2^14, showing a(n) in tiny black points if a(n) = A358971(n), else a(n) in red, and A358971(n) in blue.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%p A253288div := proc(a,n)
%p local npr,d,apr ;
%p npr := numtheory[factorset](n) ;
%p for d in npr do
%p if modp(a,d) <> 0 then
%p return false;
%p end if;
%p end do:
%p apr := numtheory[factorset](a) ;
%p if apr minus npr = {} then
%p true;
%p else
%p false;
%p end if;
%p end proc:
%p A253288 := proc(n)
%p option remember;
%p local a,i,prev,act,ev ;
%p if n =1 then
%p 1;
%p else
%p act := 1 ;
%p if type(A005361(n),'even') then
%p ev := true;
%p else
%p ev := false;
%p end if;
%p for a from 1 do
%p prev := false;
%p for i from 1 to n-1 do
%p if procname(i) = a then
%p prev := true;
%p break;
%p end if;
%p end do:
%p if not prev then
%p if A253288div(a,n) then
%p if ev or act > 1 then
%p return a;
%p else
%p act := act+1 ;
%p end if;
%p end if;
%p end if;
%p end do:
%p end if;
%p end proc:
%p seq(A253288(n),n=1..80) ; # _R. J. Mathar_, Jan 22 2015
%t nn = 1000; c[_] = False; q[_] = 1; f[n_] := f[n] = Map[Times @@ # &, Transpose@ FactorInteger[n]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimeQ@ Sqrt[n], k = Sqrt[n], SquareFreeQ[n], k = First@ f[n]; m = q[k]; While[Nand[! c[k m], k m != n, Divisible[k, First@ f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]; k *= m, True, t = 0; Set[{k, s}, {First[#], 1 + Boole@ OddQ@ Last[#]} &[f[n]]]; m = q[k]; Until[t == s, If[m > q[k], m++]; While[Nand[! c[k m], Divisible[k, First@f[m]]], m++]; t++]; If[s == 1, While[Nor[c[q[k] k], Divisible[k, First@ f[q[k]]]], q[k]++]]; k *= m]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger_, Dec 10 2022 *)
%Y Cf. A005361 (Product of exponents of prime factorization of n), A358971.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Dec 29 2014
%E Terms beyond 361 from _R. J. Mathar_, Jan 22 2015
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