A358971 -id:A358971 - cp's OEIS frontend

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

N. J. A. Sloane, Dec 29 2014

nonn

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

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

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

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

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

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 *)

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

A358916 a(1) = 1. Thereafter a(n) is the least novel k != n such that A007947(k)|n.

Original entry on oeis.org

1, 4, 9, 2, 25, 3, 49, 16, 27, 5, 121, 6, 169, 7, 45, 8, 289, 12, 361, 10, 63, 11, 529, 18, 125, 13, 81, 14, 841, 15, 961, 64, 99, 17, 175, 24, 1369, 19, 117, 20, 1681, 21, 1849, 22, 75, 23, 2209, 32, 343, 40, 153, 26, 2809, 36, 275, 28, 171, 29, 3481, 30, 3721

Offset: 1

David James Sycamore, Dec 05 2022

nonn

In other words, a(1) = 1, then for n > 1, a(n) is the least number k, not occurring earlier, whose squarefree kernel (rad(k)) is a divisor of n.
A permutation of the positive integers. - Robert Israel, Dec 11 2022
From Michael De Vlieger, Dec 06 2022, corrected by Robert Israel, Dec 11 2022: (Start)
Some consequences of definition:
Prime n = p implies a(p) = p^2, comprising maxima.
n = 2p implies a(2p) = p, n = 4p implies a(4p) = 2p.
n = 2^e with e >= 1 implies a(2^e) = 2^(e+1) if e is odd, 2^(e-1) if e is even.
n = p^e with e >= 1 and p an odd prime implies a(n) = p^(e+1).
Composite squarefree 2n implies a(2n) = n, comprising minima.
gcd(n, n +/- 1) = 1 implies gcd(a(n), a(n +/- 1)) = 1.
Let K = rad(n); a(n) is an element of R_K, the list of K-regular numbers, 1 and those whose prime divisors are restricted to p | K. For example, if K = 6, then a(n) != n is in A003586, and if K = 10, then a(n) != n is in A003592. (End)

			a(5) = 25 because rad(25) = 5  and there is no smaller number not equal to 5 which has this property.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^12, highlighting and labeling primes in red, composite prime powers in gold and labeling in orange, composite squarefree in green, numbers that are products of composite prime powers in large light blue and labeling in blue, and all other numbers in dark blue. The first 36 terms are simply labeled in black.

Cf. A000005, A000040, A001248, A005117, A007947, A032741, A358820, A358971.

N:= 100: # for a(1)..a(N)
R:= map(NumberTheory:-Radical, [$1..N^2]):
A[1]:= 1:
Agenda:= [$2..N^2]:
for n from 2 to N do
  if isprime(R[n]) then
    if R[n] = 2 and padic:-ordp(n,2)::even then A[n]:= n/2
    else A[n]:= R[n]*n
    fi;
    if A[n] <= N then Agenda:= subs(A[n]=NULL,Agenda) fi;
    next
  fi;
  found:= false;
  for j from 1 to nops(Agenda) do
    x:= Agenda[j];
    if x <> n  and n mod R[x] = 0 then
      A[n]:= x; Agenda:= subsop(j=NULL,Agenda); found:= true; break
    fi
  od;
  if not found then break fi;
od:
convert(A,list); # Robert Israel, Dec 11 2022

nn = 120; c[] = False; f[n] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; u = 2; Do[Which[PrimeQ[n], k = n^2, PrimePowerQ[n], Set[{p, k}, {f[n], 1}]; While[Nand[! c[p^k], p^k != n], k++]; k = p^k, True, k = u; While[Nand[! c[k], k != n, Divisible[n, f[k]]], k++]]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Dec 06 2022 *)

For n = p^k, where p is prime and k >= 1, a(n) = p^(k+1). In particular, a(p) = p^2 (records).

More terms from Michael De Vlieger, Dec 07 2022

A358786 a(1) = 1. For n > 1, a(n) is least novel k != n such that rad(k) = rad(n) and either k | n or n | k, where rad is A007947.

Original entry on oeis.org

1, 4, 9, 2, 25, 12, 49, 16, 3, 20, 121, 6, 169, 28, 45, 8, 289, 36, 361, 10, 63, 44, 529, 48, 5, 52, 81, 14, 841, 60, 961, 64, 99, 68, 175, 18, 1369, 76, 117, 80, 1681, 84, 1849, 22, 15, 92, 2209, 24, 7, 100, 153, 26, 2809, 108, 275, 112, 171, 116, 3481, 30, 3721

Offset: 1

Michael De Vlieger, Dec 08 2022

nonn

Variant of A358971 that additionally requires either k | n or n | k. This version eliminates nondivisor n and a(n) seen in a scatterplot of A358971. First differs from A358971 at n = 18.
Some consequences of definition:
There are no fixed points outside of a(1) = 1.
Prime power p^e implies a(p^e) = p^(e+1) for odd e, else p^(e-1). Hence a(p) = p^2 comprise maxima, while a(p^2) = p comprise minima.
Let lpf(m) = least prime factor of m. Squarefree m implies a(m) = lpf(m)*m and a(lpf(m)*m) = m, as seen in scatterplot in rays with slope p and 1/p, respectively. Therefore squarefree numbers are sequestered along or below a(n/2) = n/2.
Let K = rad(n); a(n) and n (such that a(n) != n) belong to the same sequence K*R_K, where R_K is the list of K-regular numbers, 1 and those whose prime divisors are restricted to p | K. For example, if K = 6, then a(n) and n belong to 6*A003586, and if K = 10, then a(n) and n belong to 10*A003592.

Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n) n = 1..2^10, showing primes in red, composite prime powers (in A246547) in gold, squarefree composites (in A120944) in green, numbers neither squarefree nor prime power (in A126706) in blue, highlighting numbers in A286708 in large light blue. Gold and light blue numbers are in A001694. Maxima are a(p) = p^2, minima are a(p^2) = p.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing n | a(n) in green, a(n) | n in red.

Cf. A007947, A358971.

nn = 61; c[] = False; q[] = 1; f[n_] := f[n] = Times @@ FactorInteger[n][[All, 1]]; a[1] = 1; c[1] = True; Do[Which[PrimePowerQ[n], k = If[OddQ[#2], #1^(#2 + 1), #1^(#2 - 1)] & @@ First@ FactorInteger[n], PrimeQ@ Sqrt[n], k = Sqrt[n], True, k = f[n]; m = q[k]; While[Nand[! c[k m], Or[Divisible[k m, n], Divisible[n, k m]], k m != n, Divisible[k, f[m]]], m++]; While[Nor[c[q[k] k], Divisible[k, f[q[k]]]], q[k]++]; k *= m]; Set[{a[n], c[k]}, {k, True}], {n, 2, nn}]; Array[a, nn]

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A358916 a(1) = 1. Thereafter a(n) is the least novel k != n such that A007947(k)|n.

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A358786 a(1) = 1. For n > 1, a(n) is least novel k != n such that rad(k) = rad(n) and either k | n or n | k, where rad is A007947.

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