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.

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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..61.

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.

Mathematica

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]

Crossrefs

Cf. A007947, A358971.

Sequence in context: A064505 A253288 A358971 * A360541 A365298 A367932

Adjacent sequences: A358783 A358784 A358785 * A358787 A358788 A358789

Keyword

nonn

Author

Michael De Vlieger, Dec 08 2022

Status

approved