A353916 a(1) = 1, a(2) = 2. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 2, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)).

1, 2, 6, 3, 12, 4, 10, 5, 15, 9, 18, 8, 14, 7, 21, 27, 24, 16, 20, 25, 30, 32, 22, 11, 33, 66, 36, 42, 28, 49, 35, 70, 40, 60, 45, 81, 39, 13, 26, 64, 34, 17, 51, 102, 48, 78, 52, 128, 38, 19, 57, 114, 54, 84, 56, 126, 63, 105, 75, 90, 50, 110, 44, 121, 55, 125

Offset: 1

Author

Michael De Vlieger, May 10 2022

Keywords

• nonn

Comments

Let P = the set of distinct prime divisors of j = a(n-1), and let Q = the set of distinct prime divisors of k = a(n). Let g = gcd(j, k) > 1 and let G = {P intersect Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa.

Coprimality and equality are forbidden, forcing primes into divisibility. Because of this, a(i) = mp -> a(i+1) = p -> a(i+2) = sp, with composite m and s not powers of p > 2. For n > i, multiples of p including powers may appear in the sequence. Consequently, odd primes enter the sequence late.

Numbers m that immediately precede and follow prime p have omega(m) > 1.

For any adjacent pair of terms with n > 1, if one is prime then the other cannot be a power of that prime, since such a pair would have the same number of distinct prime divisors.

This sequence requires an asymmetric version of the relation of j and k seen in A337687. In that sequence, we have P != Q, |P| > |G| and |Q| > |G|, therefore we have symmetry in that there is at least 1 prime p | j that does not divide k, and at least 1 prime q | k that does not divide j. That sequence occurs among composites m with omega(m) > 1, but this sequence admits primes, since, for |G| = 1, we must have |P| or |Q| equal to 1, and there is no prohibition for multiplicity to exceed 1.

A353917 is a version of this sequence that prohibits divisibility, hence primes do not appear, but composite prime powers do.

Open question: do the primes appear in order? (They do for n <= 2^16).

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^12, showing records in red and local minima in blue, highlighting fixed points in gold and primes in green.
• Michael De Vlieger, Prime power factor study of a(n), n = 1..2^13 where n increases from left to right, and pi(p) from bottom to top in the top image. Multiplicity e of p^e is represented by black for e = 1, red for e = 2, and a color function to the maximum multiplicity in the range thereafter. On the lower bar we show primes in red, composite prime powers in yellow, squarefree numbers m with omega(m) > 1 in green, and nonsquarefree composite numbers m with omega(m) > 1 (i.e., in A126706) in blue.
• Index entries for sequences related to EKG sequence

Crossrefs

Cf. A064413, A337687, A353917.

Programs

Mathematica

nn = 2^8; c[_] = 0; a[1] = c[1] = 1; j = a[2] = c[2] = 2; u = 3; Do[Set[k, u]; Set[m, PrimeNu[j]]; While[Nand[c[k] == 0, And[#2 > #3, #1 == #3] & @@ Append[Sort[{m, PrimeNu[k]}], PrimeNu[GCD[j, k]]]], k++]; Set[{a[i], c[k]}, {k, i}]; j = k; If[k == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn]