A309310 Intersection of A046346 and (A046346-2).

70, 286, 646, 648, 18048, 26752, 39128, 40000, 55648, 60760, 64798, 72928, 73726, 164736, 167440, 174018, 298298, 324478, 332748, 352798, 361788, 373246, 434928, 649798, 719998, 862750, 871198

Offset

1,1

Comments

Numbers k such that both k and k+2 are composite and each is divisible by the sum of its prime factors (counted with multiplicity).

There are at least two cases where k, k+2 and k+4 are all in A046346: k=646 and k=38104990. Are there more?

Up to 3*10^12 there is only one other such triple for k=590269019100. - Giovanni Resta, Jul 24 2019

Links

Amiram Eldar, Table of n, a(n) for n = 1..2000 (terms 1..200 from Robert Israel)

Example

a(1)=70 is a term because 70=2*5*7 is divisible by 2+5+7=14 and 72=2^3*3^2 is divisible by 2*3+3*2=12.

Maple

filter:= proc(n) local E, t;
  if isprime(n) then return false fi;
  E:= ifactors(n)[2];
  n mod add(t[1]*t[2], t=E) = 0
end proc:
A046346:= select(filter, {$2..10^6}):
sort(convert(A046346 intersect map(`-`, A046346, 2), list));

Mathematica

Select[Partition[Select[Range[2, 10^6], And[! PrimeQ[#], IntegerQ[#/Total[Times @@@ FactorInteger[#]]]] &], 2, 1], Subtract @@ # == -2 &][[All, 1]] (* Michael De Vlieger, Jul 22 2019 *)
cdsQ[n_]:=CompositeQ[n]&&Divisible[n, Total[Times@@@FactorInteger[n]]]; SequencePosition[Table[If[cdsQ[n], 1, 0], {n, 872000}], {1, _, 1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 23 2019 *)

Crossrefs

Cf. A046346.

Sequence in context: A245857 A227879 A072596 * A330702 A335861 A174533

Adjacent sequences: A309307 A309308 A309309 * A309311 A309312 A309313

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Jul 22 2019

Status

approved