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A145916 Even composites in A145832 with at least three distinct prime factors.

%I A145916 #7 Feb 16 2025 08:33:09
%S A145916 4346,5246,7124,9434,9698,16826,18422,18814,21826,23084,29606,30806,
%T A145916 32570,34844,35294,39614,41534,50060,52646,54164,55574,56234,63110,
%U A145916 63554,63626,64076,75206,77654,77774,80954,93716,94604,96134,99644
%N A145916 Even composites in A145832 with at least three distinct prime factors.
%C A145916 Terms of A145915 that have at least three distinct prime factors. A145915 is the sequence of even composites in A145832. A145832 is the sequence of numbers n such that for each divisor d of n, k = d + n/d is square-root smooth, i.e. p <= sqrt(k), where p is the largest prime dividing k.
%H A145916 Klaus Brockhaus, <a href="/A145916/b145916.txt">Table of n, a(n) for n=1..4000</a>
%H A145916 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RoundNumber.html">Round Number</a>
%e A145916 5246 = 2*43*61 is even and composite and has three distinct prime factors, 1, 2, 43, 61, 86, 122, 2623, 5246 are its divisors. 1+5246/1 = 5246+5246/5246 = 5247 = 3^2*11*53 and 53 < 72 < sqrt(5247); 2+5246/2 = 2623+5246/2623 = 2625 = 3*5^3*7 and 7 < 51 < sqrt(2625); 43+5246/43 = 122+5246/122 = 165 = 3*5^11 and 11 < 12 < sqrt(165); 61+5246/61 = 86+5246/86 = 147 = 3*7^2 and 7 < 12 < sqrt(147). Hence 5246 is in the sequence.
%o A145916 (Magma) [ n: n in [4..100000 by 2] | #PrimeDivisors(n) gt 2 and forall{ k: k in [ Integers()!(d+n/d): d in [ D[j]: j in [1..a] ] ] | k ge (IsEmpty(T) select 1 else Max(T) where T is [ x[1]: x in Factorization(k) ])^2 } where a is IsOdd(#D) select (#D+1)/2 else #D/2 where D is Divisors(n) ];
%Y A145916 Cf. A145832, A048098 (square-root smooth numbers), A145915.
%K A145916 nonn
%O A145916 1,1
%A A145916 _Klaus Brockhaus_, Oct 26 2008