A323640 Numbers m having at least one pair (x,y) of divisors with x

6, 20, 56, 70, 110, 182, 272, 286, 308, 506, 646, 650, 812, 884, 992, 1150, 1406, 1672, 1748, 1798, 1892, 2162, 2756, 2990, 3422, 3526, 3782, 4030, 4466, 4556, 4606, 4930, 5402, 5510, 5704, 6032, 6068, 6806, 7198, 7310, 7378, 7832, 7904, 8084, 8170, 8246, 8584, 8710

Offset: 1

Author

David A. Corneth, Aug 31 2019

Keywords

• nonn

Comments

Primitive terms of A094519.

From Bernard Schott, Aug 31 2019: (Start)

Some subsequences (this list is not exhaustive):

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, ...

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, ...

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, ...

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)

Examples

			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.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..11106 (terms <= 5*10^7)

Crossrefs

Cf. A094519.

Programs

Maple

filter:= proc(n) local D,i,j,nD;
  D:= numtheory:-divisors(n);
  nD:= nops(D);
  for i from 1 to nD-1 do
    for j from i+1 to nD do
      if (n/(D[i]+D[j]))::integer then return true fi
  od od;
  false
end proc:
N:= 10000: # for terms <= N
C:= Vector(N):
R:= NULL:
for i from 1 to N do
  if C[i]=0 and filter(i) then
    R:= R, i;
    C[[seq(i*j,j=2..N/i)]]:= 1
  fi
od:
R; # Robert Israel, Sep 02 2019

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 }
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 }