A352098 a(1) = 6, a(2) = 15; let i = a(n-2) and j = a(n-1); a(n) = least k not already in the sequence such that gcd(j, k) = 1 and gcd(i, k) = m > 1 and both omega(i) and omega(k) exceed omega(m), where omega = A001221.

6, 15, 14, 33, 10, 21, 22, 35, 12, 55, 26, 45, 28, 39, 20, 51, 38, 63, 34, 57, 40, 69, 44, 75, 46, 65, 18, 85, 52, 95, 24, 115, 56, 135, 58, 93, 50, 87, 62, 99, 68, 77, 30, 91, 66, 119, 60, 133, 74, 105, 76, 111, 70, 117, 82, 129, 80, 123, 86, 141, 88, 147, 92

Offset

1,1

Comments

Variant of A098550 analogous to A337687 regarding its relationship to A064413.

Theorem: the sequence contains numbers in A024619. Proof: with m = gcd(i, k) > 1, we have omega(m) >= 1, however, both omega(i) and omega(k) must exceed omega(m). Therefore, if no number in the sequence has a single distinct prime factor, none can arise. This restricts the sequence to numbers that are not prime powers.

Theorem: i and k are nondivisors of one another. Proof: both omega(i) and omega(k) exceed omega(m), therefore there exists some prime p | i and some prime q | k, yet, p does not divide k and q does not divide i. Hence i does not divide k and k does not divide i.

The numbers i and k have the relationship described in A272619, a sort of relationship described in A045763.

Theorem: if prime p | j then p does not divide k. Consequence of coprimality axiom gcd(j, k) = 1. Hence, even terms are nonadjacent in the sequence. Therefore we begin this sequence with {6, 15}.

Composite quasi-rays in the Yellowstone sequence scatterplot are retained, bifurcated according to parity for same reasons as in that sequence.

Conjecture: permutation of A024619.

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..2^10, showing records in red and local minima in blue, highlighting even terms in gold.

Michael De Vlieger, Prime power factor diagram for p^e | a(n), n = 1..320, where the upper portion plots p^e at (n, pi(p)), with a color function representing e where black = 1, red = 2, etc. to magenta representing the largest e in the range. The lower portion uses green to represent squarefree a(n) and blue to represent other composite a(n).

Mathematica

c[_] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {6, 15}]; Set[{i, j, u, nn}, {a[1], a[2], 10, 120}]; Do[k = u; m = PrimeNu[i]; While[Nand[c[k] == 0, And[# > 1, And[m > #, PrimeNu[k] > #] &@ PrimeNu[#]] &@ GCD[i, k], CoprimeQ[j, k]], k++]; Set[{a[n], c[k], i, j}, {k, n, j, k}]; If[k == u, While[Nand[c[u] == 0, ! PrimePowerQ[u]], u++]], {n, 3, nn}]; Array[a, nn]

Crossrefs

Cf. A001221, A024619, A045763, A064413, A098550, A337687.

Sequence in context: A265388 A334352 A128512 * A201142 A200902 A205132

Adjacent sequences: A352095 A352096 A352097 * A352099 A352100 A352101

Keyword

nonn

Author

Michael De Vlieger, Jun 03 2022

Status

approved