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%I #31 Sep 08 2022 08:45:19
%S 1,3,5,7,9,15,21,25,27,35,45,49,63,75,81,105,125,135,147,175,189,225,
%T 243,245,315,343,375,405,441,525,567,625,675,729,735,875,945,1029,
%U 1125,1215,1225,1323,1575,1701,1715,1875,2025,2187,2205,2401,2625,2835,3087
%N Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.
%C The Heinz numbers of the partitions into parts 2,3, and 4 (including the number 1, the Heinz number of the empty partition). We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by _Alois P. Heinz_ in A215366 as an "encoding" of a partition). For example, for the partition [2,3,3,4] the Heinz number is 3*5*5*7 = 525; it is in the sequence. - _Emeric Deutsch_ , May 21 2015
%C Numbers m | 105^e with integer e >= 0. - _Michael De Vlieger_, Aug 22 2019
%H Michael De Vlieger, <a href="/A108347/b108347.txt">Table of n, a(n) for n = 1..10000</a>
%H Vaclav Kotesovec, <a href="/A108347/a108347.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F Sum_{n>=1} 1/a(n) = (3*5*7)/((3-1)*(5-1)*(7-1)) = 35/16. - _Amiram Eldar_, Sep 22 2020
%F a(n) ~ exp((6*log(3)*log(5)*log(7)*n)^(1/3)) / sqrt(105). - _Vaclav Kotesovec_, Sep 23 2020
%p with(numtheory): S := {}: for j to 3100 do if `subset`(factorset(j), {3, 5, 7}) then S := `union`(S, {j}) else end if end do: S; # _Emeric Deutsch_, May 21 2015
%t With[{n = 3087}, Sort@ Flatten@ Table[3^i * 5^j * 7^k, {i, 0, Log[3, n]}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(3^i*5^j)]}]] (* _Michael De Vlieger_, Aug 22 2019 *)
%o (Magma) [n: n in [1..4000] | PrimeDivisors(n) subset [3,5,7]]; // _Bruno Berselli_, Sep 24 2012
%Y Cf. A003586, A003591-A003595, A051037, A108319, A108513, A215366.
%K nonn
%O 1,2
%A Douglas Winston (douglas.winston(AT)srupc.com), Jul 01 2005
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