A336443 -id:A336443 - cp's OEIS frontend

A333966 Positive integers where the number of triples of divisors (d1, d2, d3) such that d1 < d2 < d3 < 2*d1 and each pair of these divisors is pairwise coprime, sets a new record.

1, 60, 280, 420, 840, 1260, 2520, 6930, 9240, 13860, 27720, 55440, 60060, 120120, 180180, 240240, 360360, 720720, 1021020, 1801800, 2042040, 2282280, 2762760, 3063060, 4084080, 4564560, 6126120, 12252240, 19399380, 24504480, 30630600, 36756720, 38798760, 58198140, 77597520

Offset: 1

David A. Corneth, Jul 22 2020

nonn

Records are 0, 1, 2, 3, 4, 5, 8, 9, 11, 13, 19, ...
Are terms > 4564560 products of primorials (cf. A025487)? Terms 4564560 < k <= 54765047434897800 are.
In a triple (d1, d2, d3) such that lcm(d1, d2, d3) = d1*d2*d2 <= k we must have d1^3 < k. Proof: Suppose d1^3 >= n. Then d1 * d2 * d3 > n since d2 > d1 and d3 > d1.   Since any pair is coprime  d1 * d2 * d3  = LCM(d1,d2,d3) is a divisor of n. A contradiction. - David A. Corneth and Amiram Eldar, Jul 28 2020

			280 has two such divisor triples; (4, 5, 7) and (5, 7, 8) and no number less than 280 has at least two such triples so 280 is in the sequence.

David A. Corneth, records; number of such triples in divisors of a(n)

Cf. A336442, A336443, A336628, A336629.

upto(n) = { v = vectorsmall(n); for(i = 2, sqrtnint(n, 3), for(j = i + 1, min(sqrtint(n \ i), 2*i-2), g = gcd(i, j); if(g == 1, l = i * j / g; for(k = j + 1, min(2*i-1, n \ (i*j)), if(gcd(l, k) == 1, p = l*k; forstep(m = p, n, p, v[m]++ ); t++ ))))); my(res=List(1), r=v[1]); for(i=2, #v, if(v[i]>r, r=v[i]; listput(res,i))); res }

A336442 Numbers having 3 pairwise coprime divisors, {d_1, d_2, d_3}, such that d_1 < d_2 < d_3 < 2*d_1.

Original entry on oeis.org

60, 120, 140, 180, 210, 240, 280, 300, 315, 360, 420, 462, 480, 504, 540, 560, 600, 616, 630, 660, 693, 700, 720, 728, 770, 780, 792, 819, 840, 900, 910, 924, 936, 945, 960, 980, 990, 1001, 1008, 1020, 1050, 1080, 1092, 1120, 1140, 1144, 1170, 1200, 1232, 1260

Offset: 1

Amiram Eldar, Jul 21 2020

nonn

Erdős (1970) proved that the asymptotic density of this sequence exists and is less than 1.
The numbers of terms not exceeding 10^k for k = 1, 2, ... are 0, 1, 37, 543, 6529, 73578, 798916, 8480417, 88832422, ...
Any positive multiple of any term of this sequence is also a term. The primitive terms are in A336443.

			60 is a term since {3, 4, 5} are divisors of 60, gcd(3,4) = gcd(4,5) = gcd(3,5) = 1 and 3 < 4 < 5 < 2*3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, Some extremal problems in combinatorial number theory, in the book Hari Shankar (ed.), Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio (1970), pp. 123-133.

Subsequence of A005279.

A336443 is a subsequence.

divQ[n_] := AnyTrue[Subsets[Divisors[n], {3}], And @@ CoprimeQ @@@ Subsets[#, {2}] && #[[3]] < 2 * #[[1]] &]; Select[Range[1500], divQ]

cp's OEIS Frontend

A333966 Positive integers where the number of triples of divisors (d1, d2, d3) such that d1 < d2 < d3 < 2*d1 and each pair of these divisors is pairwise coprime, sets a new record.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI