A283423 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.

2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 64, 100, 128, 162, 256, 272, 294, 342, 486, 500, 512, 1024, 1458, 1806, 2048, 2058, 2500, 4096, 4374, 4624, 6498, 8192, 10100, 12500, 13122, 14406, 16384, 23994, 26406, 32768, 34362, 39366, 47058

Offset

1,1

Comments

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

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).

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).

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

Links

Giovanni Resta, Table of n, a(n) for n = 1..131 (terms < 10^11, first 101 terms from Amiram Eldar)

John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.

Example

m = 18 is in the sequence because 1/18 + 1/2 + 1/3 + 1/9 = 1.
m = 12 is NOT in the sequence because 1/12 + 1/2 + 1/4 + 1/3 != 1.

Mathematica

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 *)

Crossrefs

Cf. A005835, A054377, A073935.

Sequence in context: A333020 A325792 A325780 * A073935 A325764 A258119

Adjacent sequences: A283420 A283421 A283422 * A283424 A283425 A283426

Keyword

nonn

Author

John Machacek, May 27 2017

Status

approved