A343597 -id:A343597 - cp's OEIS frontend

A080672 Numbers having divisors 2 or 3 or 5 or 7.

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Cino Hilliard, Mar 02 2003

A020639(a(n)) <= 7; A210679(a(n)) > 0. - Reinhard Zumkeller, Apr 02 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 162.

Cf. A020639, A008364 (complement).

Subsequences: A002473, A343597.

a080672 n = a080672_list !! (n-1)
a080672_list = filter ((<= 7) . a020639) [2..]
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],Length[Intersection[Divisors[#],{2,3,5,7}]]>0&] (* Harvey P. Dale, Apr 03 2024 *)

div2357(n)= for(x=1,n, if(gcd(x,210)<>1,print1(x" ")) )

is(n)=gcd(n,210)>1 \\ Charles R Greathouse IV, Sep 14 2015

From Charles R Greathouse IV, Sep 14 2015: (Start)
a(n) = 35n/27 + O(1).
For n > 162, a(n) = a(n-162) + 210. [Corrected by Peter Munn, Apr 22 2021]
(End)
For n < 162, a(n) = 210 - a(162-n). - Peter Munn, Apr 22 2021

Offset fixed by Reinhard Zumkeller, Apr 02 2012

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

Original entry on oeis.org

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

Peter Munn, Apr 21 2021

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.

			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, ... .

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

Cf. A002110, A002473, A281891, A343597.

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

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

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A080672 Numbers having divisors 2 or 3 or 5 or 7.

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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.

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