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A108347 Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.

%I A108347 #42 May 22 2025 20:56:16
%S A108347 1,3,5,7,9,15,21,25,27,35,45,49,63,75,81,105,125,135,147,175,189,225,
%T A108347 243,245,315,343,375,405,441,525,567,625,675,729,735,875,945,1029,
%U A108347 1125,1215,1225,1323,1575,1701,1715,1875,2025,2187,2205,2401,2625,2835,3087
%N A108347 Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.
%C A108347 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 A108347 Numbers m | 105^e with integer e >= 0. - _Michael De Vlieger_, Aug 22 2019
%H A108347 Michael De Vlieger, <a href="/A108347/b108347.txt">Table of n, a(n) for n = 1..10000</a>
%H A108347 Vaclav Kotesovec, <a href="/A108347/a108347.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A108347 Sum_{n>=1} 1/a(n) = (3*5*7)/((3-1)*(5-1)*(7-1)) = 35/16. - _Amiram Eldar_, Sep 22 2020
%F A108347 a(n) ~ exp((6*log(3)*log(5)*log(7)*n)^(1/3)) / sqrt(105). - _Vaclav Kotesovec_, Sep 23 2020
%p A108347 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
%p A108347 # alternative
%p A108347 isA108347 := proc(n)
%p A108347       if n = 1 then
%p A108347         true;
%p A108347     else
%p A108347         return (numtheory[factorset](n) minus {3, 5, 7} = {} );
%p A108347     end if;
%p A108347 end proc:
%p A108347 A108347 := proc(n)
%p A108347      option remember;
%p A108347      if n = 1 then
%p A108347         1;
%p A108347     else
%p A108347         for a from procname(n-1)+1 do
%p A108347             if isA108347(a) then
%p A108347                 return a;
%p A108347             end if;
%p A108347         end do:
%p A108347     end if;
%p A108347 end proc:
%p A108347 seq(A108347(n),n=1..80); # _R. J. Mathar_, Jun 06 2024
%t A108347 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 A108347 (Magma) [n: n in [1..4000] | PrimeDivisors(n) subset [3,5,7]]; // _Bruno Berselli_, Sep 24 2012
%o A108347 (Python)
%o A108347 from sympy import integer_log
%o A108347 def A108347(n):
%o A108347     def bisection(f,kmin=0,kmax=1):
%o A108347         while f(kmax) > kmax: kmax <<= 1
%o A108347         while kmax-kmin > 1:
%o A108347             kmid = kmax+kmin>>1
%o A108347             if f(kmid) <= kmid:
%o A108347                 kmax = kmid
%o A108347             else:
%o A108347                 kmin = kmid
%o A108347         return kmax
%o A108347     def f(x):
%o A108347         c = n+x
%o A108347         for i in range(integer_log(x,7)[0]+1):
%o A108347             for j in range(integer_log(m:=x//7**i,5)[0]+1):
%o A108347                 c -= integer_log(m//5**j,3)[0]+1
%o A108347         return c
%o A108347     return bisection(f,n,n) # _Chai Wah Wu_, Sep 16 2024
%Y A108347 Cf. A003586, A003591-A003595, A051037, A108319, A108513, A215366.
%Y A108347 The odd terms of A002473.
%K A108347 nonn
%O A108347 1,2
%A A108347 Douglas Winston (douglas.winston(AT)srupc.com), Jul 01 2005