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A343598 Positive integers k such that exactly half the integers in [1..k] are divisible by a 7-smooth composite number.

%I A343598 #16 Feb 16 2025 08:34:02
%S A343598 10,12,14,62,74,86,88,90,92,100,102,104,106,108,110,112,114,116,118,
%T A343598 120,126,128,130,132,136,138,140,154,156,172,174,178,180,182,184,186,
%U A343598 188,194,202,204,208,210,212,246,248,250,252,256,258,260,262,264,266,268,270
%N A343598 Positive integers k such that exactly half the integers in [1..k] are divisible by a 7-smooth composite number.
%C A343598 In every interval of 44100 integers, exactly 22164 are divisible by a 7-smooth composite number. 44100 = (2*3*5*7)^2 = A002110(4)^2 and 22164 = A281891(4,2). See A281891 for more details.
%C A343598 The sequence is finite with largest term a(136) = 1406.
%H A343598 Jinyuan Wang, <a href="/A343598/b343598.txt">Table of n, a(n) for n = 1..136</a>
%H A343598 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth Number</a>.
%F A343598 {a(n)} = {k : k = 2*m, A343597(m) <= k < A343597(m + 1)}.
%e A343598 The numbers divisible by a 7-smooth composite number are given in A343597. List in a row the numbers that are present, with the absent numbers (aligned) in a row below. Where the count of absent numbers matches that of those present, draw a vertical line, such that all the numbers to the left are less than all the numbers to the right. See the figure below, where the rows are segmented for practical reasons:
%e A343598 --------------
%e A343598 Present :   4   6   8   9  10 | 12 | 14 | 15  16  18  20  21
%e A343598 Missing :   1   2   3   5   7 | 11 | 13 | 17  19  22  23  26
%e A343598 ----------
%e A343598 Present :  24  25  27  28  30  32  35  36  40  42  44  45  48
%e A343598 Missing :  29  31  33  34  37  38  39  41  43  46  47  51  53
%e A343598 ----------
%e A343598 Present :  49  50  52  54  56  60 | 63  64  66  68  70  72 |
%e A343598 Missing :  55  57  58  59  61  62 | 65  67  69  71  73  74 |
%e A343598 ----------
%e A343598   ...
%e A343598 --------------
%e A343598 Listing the largest number to the left of each vertical line gives this sequence: 10, 12, 14, 62, 74, ... .
%o A343598 (PARI) upto(n) = { my(t = 0, res = List()); for(i = 1, n, if(isdivby(i), t++; ); if(2*t == i, listput(res, i))); res }
%o A343598 isdivby(n) = { my(v = [4, 6, 9, 10, 14, 15, 21, 25, 35, 49]); for(i = 1, #v, if(n%v[i] == 0, return(1))); 0 } \\ _David A. Corneth_, Apr 24 2021
%Y A343598 Cf. A002110, A002473, A281891, A343597.
%K A343598 nonn,easy,fini,full
%O A343598 1,1
%A A343598 _Peter Munn_, Apr 21 2021