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

1,1

Links

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

Example

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.

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 *)

Crossrefs

Cf. A091191, A283418.

Sequence in context: A229326 A365107 A008528 * A020881 A285043 A223241

Adjacent sequences: A373313 A373314 A373315 * A373317 A373318 A373319

Keyword

nonn

Author

Abhiram R Devesh, May 31 2024

Status

approved