A343597 - OEIS

%I #16 Feb 16 2025 08:34:02

%S 4,6,8,9,10,12,14,15,16,18,20,21,24,25,27,28,30,32,35,36,40,42,44,45,

%T 48,49,50,52,54,56,60,63,64,66,68,70,72,75,76,78,80,81,84,88,90,92,96,

%U 98,99,100,102,104,105,108,110,112,114,116,117,120,124,125,126,128,130

%N Numbers divisible by a 7-smooth composite number.

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

%C 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.

%C 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.

%H Amiram Eldar, <a href="/A343597/b343597.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.

%F {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.

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

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

%e 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.

%e 52 = 13 * 2 * 2 is divisible by 4, which is composite and 7-smooth, so 52 is in the sequence.

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

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

%Y Subsequence of A080672.

%K nonn,easy,changed

%O 1,1

%A _Peter Munn_, Apr 21 2021