A343597 Numbers divisible by a 7-smooth composite number.

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130

Offset

1,1

Comments

Numbers divisible by at least one of 4, 6, 9, 10, 14, 15, 21, 25, 35, 49.

Exactly half of the first 10, first 100 and first 600 positive integers are divisible by a 7-smooth composite number; the largest 7-smooth divisor of the remaining numbers is 1, 2, 3, 5 or 7.

Intervals extending to hundreds of integers with exactly 50% membership of this sequence are far from rare, some notable examples being [3000, 3999], [8000, 8999], [20000, 20999], [21000, 21999] and [23000, 23999]. This reflects the asymptotic density of the corresponding set being close to 0.5, precisely 1847 / 3675 = 0.50258503... (and membership of the set has a periodic pattern). See A343598 for further information.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Smooth Number.

Formula

{a(n)} = {k : k >= 1, 2 <= A014673(k) <= 7}, where A014673(k) = lpf(k/lpf(k)), where lpf(m) = A020639(m), the least prime factor of m.

For n >= 1, a(22164 + n) = 44100 + a(n).

For n < 22164, a(22164 - n) = 44100 - a(n).

Example

33 = 11 * 3 has divisors 1, 3, 11, 33, of which only 33 is composite. 33 is not 7-smooth, as its prime factors include 11, which is greater than 7. So 33 is not in the sequence.
52 = 13 * 2 * 2 is divisible by 4, which is composite and 7-smooth, so 52 is in the sequence.

Mathematica

Select[Range[130], Plus @@ IntegerExponent[#, {2, 3, 5, 7}] > 1 &] (* Amiram Eldar, May 04 2021 *)

Crossrefs

Cf. A002473 (7-smooth numbers), A014673, A020639, A210615 (smallest smoothest composite divisor), A343598.

Subsequence of A080672.

Sequence in context: A205783 A077094 A210441 * A199009 A070807 A246282

Adjacent sequences: A343594 A343595 A343596 * A343598 A343599 A343600

Keyword

nonn,easy

Author

Peter Munn, Apr 21 2021

Status

approved