A333966 -id:A333966 - cp's OEIS frontend

A336628 Numbers k that have 3 divisors d1, d2, d3 such that d1 < d2 < d3 < 2d1 and are pairwise coprime and d1d2*d3 = k.

60, 140, 210, 280, 315, 360, 462, 504, 616, 630, 693, 728, 770, 792, 819, 910, 924, 936, 990, 1001, 1092, 1144, 1170, 1287, 1320, 1386, 1430, 1530, 1560, 1584, 1638, 1683, 1716, 1870, 1872, 1989, 2002, 2090, 2142, 2145, 2210, 2244, 2288, 2310, 2431, 2448, 2470, 2508

Offset: 1

David A. Corneth and Amiram Eldar, Jul 28 2020

nonn

(k/4)^(1/3) < d1 < k^(1/3). Proof: as k = d1 * d2 * d3 < d1 * (2*d1) * (2*d1) = 4*d1^3 we have (k/4)^(1/3) < d1 and as k = d1 * d2 * d3 > d1 * d1 * d1 = d1^3 we have k^(1/3) > d1. Q.e.d.

			210 is in the sequence because 5*6*7 = 210 and each of these factors are pairwise coprime and 5 < 6 < 7 < 2*5 = 10.

Cf. A333966, A336629.

A336629 a(n) is the least positive integer k such that it has exactly n triples of divisors (d1, d2, d3) such that they are pairwise coprime and d1 < d2 < d3 < 2*d1.

Original entry on oeis.org

1, 60, 7140, 60060, 251940, 360360, 1369368, 1225224, 1531530, 7873866, 17687670, 5819814, 17160990, 11085360, 11741730, 19399380, 65564070, 9699690, 99533742, 85804950, 40562340, 90485220, 358888530, 504894390, 634956630, 531990690, 397687290, 512942430, 455885430, 514083570

Offset: 0

David A. Corneth, Jul 28 2020

nonn

Can we prove m is a divisor for all terms a(n) for n > N for some n? For example, are all terms from a(1) onwards divisible by 2?

For n > 0, it seems that 6|a(n) and a(n) is a Zumkeller number (A083207). Verified for n up to and including 29. - Ivan N. Ianakiev, Aug 02 2020

			a(3) = 60060 as 60060 = 28 * 39 * 55 = 33 * 35 * 52 = 35 * 39 * 44 and no positive integer < 60060 has exactly 3 such triples.

Cf. A333966, A336628.

cp's OEIS Frontend

A336628 Numbers k that have 3 divisors d1, d2, d3 such that d1 < d2 < d3 < 2d1 and are pairwise coprime and d1d2*d3 = k.

Views

Author

Keywords

Comments

Examples

Crossrefs

A336629 a(n) is the least positive integer k such that it has exactly n triples of divisors (d1, d2, d3) such that they are pairwise coprime and d1 < d2 < d3 < 2*d1.

Views

Author

Keywords

Comments

Examples

Crossrefs

cp's OEIS Frontend

A336628 Numbers k that have 3 divisors d1, d2, d3 such that d1 < d2 < d3 < 2*d1 and are pairwise coprime and d1*d2*d3 = k.

Views

Author

Keywords

Comments

Examples

Crossrefs

A336629 a(n) is the least positive integer k such that it has exactly n triples of divisors (d1, d2, d3) such that they are pairwise coprime and d1 < d2 < d3 < 2*d1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A336628 Numbers k that have 3 divisors d1, d2, d3 such that d1 < d2 < d3 < 2d1 and are pairwise coprime and d1d2*d3 = k.