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.

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

1,1

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)

Links

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

Example

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.

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

Prog

(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 }

Crossrefs

Cf. A094519.

Sequence in context: A152959 A328681 A109903 * A249406 A220020 A201149

Adjacent sequences: A323637 A323638 A323639 * A323641 A323642 A323643

Keyword

nonn

Author

David A. Corneth, Aug 31 2019

Status

approved