A373316 Numbers k such that k and k+2 are both primitive abundant numbers.

18, 102, 364, 366, 474, 532, 642, 834, 1036, 1146, 1182, 1374, 1504, 1696, 1876, 1986, 2210, 2584, 2994, 3052, 3126, 3556, 4396, 4542, 4564, 5032, 5514, 5572, 5574, 5622, 6232, 6412, 6522, 6976, 7026, 7206, 7912, 7924, 8202, 8596, 8706, 9654, 9714

Offset: 1

Author

Abhiram R Devesh, May 31 2024

Keywords

• nonn

Examples

			18 = 2*3*3 is an abundant number, but its proper divisors are 1, 2, 3, 6 and 9, none of which are abundant.
18 + 2 = 20 = 2*2*5 is an abundant number, but its proper divisors are 1, 2, 4, 5 and 10, none of which are abundant.
Thus, both 18 and 20 are primitive abundant numbers, so 18 is in the sequence.
		

Links

• Abhiram R Devesh, Table of n, a(n) for n = 1..208

Crossrefs

Cf. A091191, A283418.

Programs

Mathematica

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1); primAbQ[n_] := primAbQ[n] = (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2; Select[Range[2, 10^4], primAbQ[#] && primAbQ[# + 2] &] (* Amiram Eldar, Jul 20 2024 *)