A283423 - OEIS

%I #49 Mar 11 2024 01:50:25

%S 2,4,6,8,16,18,20,32,42,54,64,100,128,162,256,272,294,342,486,500,512,

%T 1024,1458,1806,2048,2058,2500,4096,4374,4624,6498,8192,10100,12500,

%U 13122,14406,16384,23994,26406,32768,34362,39366,47058

%N Prime power pseudoperfect numbers: numbers m > 1 such that 1/m + Sum 1/p^k = 1, where the sum is over the prime powers p^k | m.

%C Since primary pseudoperfect numbers (A054377) must be squarefree, it follows that primary pseudoperfect numbers are contained in this sequence.

%C This sequence contains all powers of 2. With the exception of the powers of 2, every prime power pseudoperfect number is a pseudoperfect number (A005835).

%C Every number in A073935 is a prime power pseudoperfect number (note: this sequence and A073935 agree for many terms but eventually differ starting at 23994 the 38th term of this sequence).

%C The number 2^k(2^k+1) is the sequence whenever 2^k+1 is a Fermat prime (A019434).

%H Giovanni Resta, <a href="/A283423/b283423.txt">Table of n, a(n) for n = 1..131</a> (terms < 10^11, first 101 terms from Amiram Eldar)

%H John Machacek, <a href="https://arxiv.org/abs/1706.01008">Egyptian Fractions and Prime Power Divisors</a>, arXiv:1706.01008 [math.NT], 2017.

%e m = 18 is in the sequence because 1/18 + 1/2 + 1/3 + 1/9 = 1.

%e m = 12 is NOT in the sequence because 1/12 + 1/2 + 1/4 + 1/3 != 1.

%t ok[n_] := Total[n/Flatten@ Table[e[[1]] ^ Range[e[[2]]], {e, FactorInteger[n]}]] + 1 == n; Select[ Range[10^5], ok] (* _Giovanni Resta_, May 27 2017 *)

%Y Cf. A005835, A054377, A073935.

%K nonn

%O 1,1

%A _John Machacek_, May 27 2017