A336442 -id:A336442 - cp's OEIS frontend

A336443 Primitive terms of A336442: terms k of A336442 such that none of the proper divisors of k are in A336442.

60, 140, 210, 315, 462, 504, 616, 693, 728, 770, 792, 819, 910, 936, 990, 1001, 1092, 1144, 1170, 1287, 1430, 1530, 1683, 1716, 1870, 1989, 2090, 2142, 2145, 2210, 2244, 2431, 2448, 2470, 2508, 2618, 2652, 2717, 2805, 2926, 2964, 2992, 3094, 3135, 3230, 3315

Offset: 1

Amiram Eldar, Jul 21 2020

nonn

Any term of A336442 is a multiple of at least one term of this sequence.

			60 is a term since it is a term of A336442 but none of its proper divisors are in A336442.
120 is not a term: although it is in A336442, it is a multiple of 60 which is also a term of A336442.

David A. Corneth, Table of n, a(n) for n = 1..10347

Analogous to A302022 as A336442 is analogous to A005279.

divQ[n_] := AnyTrue[Subsets[Divisors[n], {3}], And @@ CoprimeQ @@@ Subsets[#, {2}] && #[[3]] < 2 #[[1]] &]; primQ[n_] := divQ[n] && AllTrue[Most[Divisors[n]], ! divQ[#] &]; Select[Range[3333], primQ]

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.

Original entry on oeis.org

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 }

cp's OEIS Frontend

A336443 Primitive terms of A336442: terms k of A336442 such that none of the proper divisors of k are in A336442.

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