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A343597
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Numbers divisible by a 7-smooth composite number.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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{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).
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EXAMPLE
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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.
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MATHEMATICA
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Select[Range[130], Plus @@ IntegerExponent[#, {2, 3, 5, 7}] > 1 &] (* Amiram Eldar, May 04 2021 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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