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

Author

Michael De Vlieger, Dec 08 2022

Keywords

• nonn

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

• 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.

Crossrefs

Cf. A007947, A358971.

Programs

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]