A336628 -id:A336628 - 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 }

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

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.

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