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

10, 12, 14, 62, 74, 86, 88, 90, 92, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 126, 128, 130, 132, 136, 138, 140, 154, 156, 172, 174, 178, 180, 182, 184, 186, 188, 194, 202, 204, 208, 210, 212, 246, 248, 250, 252, 256, 258, 260, 262, 264, 266, 268, 270

Offset: 1

Author

Peter Munn, Apr 21 2021

Keywords

• nonn
• easy
• fini
• full

Comments

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.

The sequence is finite with largest term a(136) = 1406.

Examples

			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:
--------------
Present :   4   6   8   9  10 | 12 | 14 | 15  16  18  20  21
Missing :   1   2   3   5   7 | 11 | 13 | 17  19  22  23  26
----------
Present :  24  25  27  28  30  32  35  36  40  42  44  45  48
Missing :  29  31  33  34  37  38  39  41  43  46  47  51  53
----------
Present :  49  50  52  54  56  60 | 63  64  66  68  70  72 |
Missing :  55  57  58  59  61  62 | 65  67  69  71  73  74 |
----------
  ...
--------------
Listing the largest number to the left of each vertical line gives this sequence: 10, 12, 14, 62, 74, ... .
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..136
• Eric Weisstein's World of Mathematics, Smooth Number.

Crossrefs

Cf. A002110, A002473, A281891, A343597.

Programs

PARI

upto(n) = { my(t = 0, res = List()); for(i = 1, n, if(isdivby(i), t++; ); if(2*t == i, listput(res, i))); res }
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

Formula

{a(n)} = {k : k = 2*m, A343597(m) <= k < A343597(m + 1)}.