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A323640 Numbers m having at least one pair (x,y) of divisors with x such that x+y is also a divisor of m but no proper divisor of m has this property.

%I A323640 #33 Sep 14 2019 16:44:07
%S A323640 6,20,56,70,110,182,272,286,308,506,646,650,812,884,992,1150,1406,
%T A323640 1672,1748,1798,1892,2162,2756,2990,3422,3526,3782,4030,4466,4556,
%U A323640 4606,4930,5402,5510,5704,6032,6068,6806,7198,7310,7378,7832,7904,8084,8170,8246,8584,8710
%N A323640 Numbers m having at least one pair (x,y) of divisors with x<y such that x+y is also a divisor of m but no proper divisor of m has this property.
%C A323640 Primitive terms of A094519.
%C A323640 From _Bernard Schott_, Aug 31 2019: (Start)
%C A323640 Some subsequences (this list is not exhaustive):
%C A323640 1) Oblong numbers of the form (3*k+1)*(3*k+2). These are in A001504 and the pair (x,y) = (1,3*k+1). Only 6 is oblong and not of this form. The first few terms are 20, 56, 110, 182, 272, ...
%C A323640 2) Numbers of the form 2*p*q  where (p, q) is a twin prime pair. These terms are precisely A071142 \ {30} and the pair (x,y) = (2,p). The first few terms are 70, 286, 646, ...
%C A323640 3) Numbers of the form 2^2 * p * q where p and q = p+4 are primes and p > 3. These primes p are in A023200 \ {3} and the pair (x,y) = (4,p). The first few terms are 308, 884, ...
%C A323640 4) More generally, numbers of the form 2^k * p * q where p and q = p+2^k are primes and the pair (x,y) = (2^k,p). For k = 3, the smallest such term is 1672 with p = 11. (End)
%H A323640 David A. Corneth, <a href="/A323640/b323640.txt">Table of n, a(n) for n = 1..11106</a> (terms <= 5*10^7)
%e A323640 56 is in the sequence as 1, 7 and 1 + 7 = 8 are divisors of 56 and no divisor of 56 is in the sequence.
%p A323640 filter:= proc(n) local D,i,j,nD;
%p A323640   D:= numtheory:-divisors(n);
%p A323640   nD:= nops(D);
%p A323640   for i from 1 to nD-1 do
%p A323640     for j from i+1 to nD do
%p A323640       if (n/(D[i]+D[j]))::integer then return true fi
%p A323640   od od;
%p A323640   false
%p A323640 end proc:
%p A323640 N:= 10000: # for terms <= N
%p A323640 C:= Vector(N):
%p A323640 R:= NULL:
%p A323640 for i from 1 to N do
%p A323640   if C[i]=0 and filter(i) then
%p A323640     R:= R, i;
%p A323640     C[[seq(i*j,j=2..N/i)]]:= 1
%p A323640   fi
%p A323640 od:
%p A323640 R; # _Robert Israel_, Sep 02 2019
%o A323640 (PARI) upto(n) = {my(charprim = vector(n, i, 1), res = List()); for(i = 1, n, if(charprim[i] == 1, if(isA094519(i), listput(res, i); for(k = 2, n \ i, charprim[i*k] = 0 ) , charprim[i] = 0; ) ) ); res }
%o A323640 isA094519(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 }
%Y A323640 Cf. A094519.
%K A323640 nonn
%O A323640 1,1
%A A323640 _David A. Corneth_, Aug 31 2019