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A094519 Numbers having at least one pair (x,y) of divisors with x such that x+y is also a divisor.

%I A094519 #49 Apr 20 2025 02:35:34
%S A094519 6,12,18,20,24,30,36,40,42,48,54,56,60,66,70,72,78,80,84,90,96,100,
%T A094519 102,108,110,112,114,120,126,132,138,140,144,150,156,160,162,168,174,
%U A094519 180,182,186,192,198,200,204,210,216,220,222,224,228,234,240,246
%N A094519 Numbers having at least one pair (x,y) of divisors with x<y such that x+y is also a divisor.
%C A094519 If m is in the sequence then so is k*m for k > 0. Furthermore, all terms are even. - _David A. Corneth_, Aug 31 2019
%C A094519 If (x,y) = (1,m) with m > 1, then oblong numbers m*(m+1) >= 6 belong to this sequence, and each oblong number >= 6 is a primitive term of the subsequence {k*m*(m+1), k >= 1}. Examples: with pair (1,2), we get multiples of 6 (see A008588); with (1,3) we get multiples of 12 (see A008594); with (1,4) we get multiples of 20 (see A008602); with (1,7) we get multiples of 56. - _Bernard Schott_, Aug 31 2019
%C A094519 The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 22, 230, 2317, 23201, 232209, 2322920, 23232166, 232332309, 2323370184, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23233... . - _Amiram Eldar_, Apr 20 2025
%H A094519 Amiram Eldar, <a href="/A094519/b094519.txt">Table of n, a(n) for n = 1..10000</a>
%F A094519 A094518(a(n)) > 0.
%t A094519 aQ[n_] := AnyTrue[Total /@ Subsets[Divisors[n], {2}], Divisible[n, #] &]; Select[Range[250], aQ] (* _Amiram Eldar_, Aug 31 2019 *)
%o A094519 (PARI) is(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 } \\ _David A. Corneth_, Aug 31 2019
%o A094519 (Python)
%o A094519 from itertools import count, islice
%o A094519 from sympy import divisors
%o A094519 def A094519_gen(): # generator of terms
%o A094519     for n in count(1):
%o A094519         for i in range(1,len(d:=divisors(n))):
%o A094519             di = d[i]
%o A094519             for j in range(i):
%o A094519                 if n % (di+d[j]) == 0:
%o A094519                     yield n
%o A094519                     break
%o A094519             else:
%o A094519                 continue
%o A094519             break
%o A094519 A094519_list = list(islice(A094519_gen(),20)) # _Chai Wah Wu_, Dec 26 2021
%Y A094519 Cf. A094518.
%Y A094519 Complement of A094520.
%Y A094519 A superset of A088723. - _R. J. Mathar_, Sep 16 2007
%Y A094519 Subsequences: A002378 \ {0, 2}, A008588 \ {0}, A008602 \ {0}.
%K A094519 nonn
%O A094519 1,1
%A A094519 _Reinhard Zumkeller_, May 06 2004