A253288 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.

1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 18, 361, 10, 63, 44, 529, 36, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 24, 1369, 76, 117, 50, 1681, 84, 1849, 22, 15, 92, 2209, 48, 7, 40, 153, 26, 2809, 72, 275, 98, 171, 116, 3481, 30, 3721, 124, 21, 32

Offset: 1

Author

N. J. A. Sloane, Dec 29 2014

Keywords

• nonn

Comments

This sequence is permutation of the positive integers.

The prime p occurs at n = p^2.

Multiples of a number x have density 1/x.

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

References

• Brad Klee, Posting to Sequence Fans Mailing List, Dec 21, 2014.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Log log scatterplot of a(n) <= 12000, 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).
• Michael De Vlieger, Log log scatterplot of a(n) <= 2^14, 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.
• Michael De Vlieger, Log log scatterplot of a(n) <= 80000, 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.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A005361 (Product of exponents of prime factorization of n), A358971.

Programs

Maple

A253288div := proc(a,n)
    local npr,d,apr ;
    npr := numtheory[factorset](n) ;
    for d in npr do
        if modp(a,d) <> 0 then
            return false;
        end if;
    end do:
    apr := numtheory[factorset](a) ;
    if apr minus npr = {} then
        true;
    else
        false;
    end if;
end proc:
A253288 := proc(n)
    option remember;
    local a,i,prev,act,ev ;
    if n =1 then
        1;
    else
        act := 1 ;
        if type(A005361(n),'even') then
            ev := true;
        else
            ev := false;
        end if;
        for a from 1 do
            prev := false;
            for i from 1 to n-1 do
                if procname(i) = a then
                    prev := true;
                    break;
                end if;
            end do:
            if not prev then
                if A253288div(a,n) then
                    if ev or act > 1 then
                        return a;
                    else
                        act := act+1 ;
                    end if;
                end if;
            end if;
        end do:
    end if;
end proc:
seq(A253288(n),n=1..80) ; # R. J. Mathar, Jan 22 2015

Mathematica

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 *)

Extensions

Terms beyond 361 from R. J. Mathar, Jan 22 2015