login
A286497
Prime power Giuga numbers: composite numbers n > 1 such that -1/n + sum 1/p^k = 1, where the sum is over the prime powers p^k dividing n.
0
12, 30, 56, 306, 380, 858, 992, 1722, 2552, 2862, 16256, 30704, 66198, 73712, 86142, 249500, 629802, 1703872, 6127552, 16191736, 19127502, 35359900, 67100672, 101999900, 172173762, 182552538, 266677578, 575688042, 1180712682, 2214408306, 6179139056, 17179738112, 21083999500
OFFSET
1,1
COMMENTS
Since Giuga numbers (A007850) must be squarefree, it follows all Giuga numbers are contained in this sequence.
The number 2^k (2^k-1) is in this sequence whenever 2^k-1 is a Mersenne prime (A000668).
LINKS
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.
EXAMPLE
n = 12 is in the sequence because -1/12 + 1/2 + 1/2^2 + 1/3 = 1.
n = 18 is NOT in the sequence because -1/18 + 1/2 + 1/3 + 1/3^2 != 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
Sequence in context: A080385 A120090 A280344 * A086830 A084699 A323441
KEYWORD
nonn
AUTHOR
John Machacek, May 27 2017
EXTENSIONS
a(20)-a(31) from Giovanni Resta, May 27 2017
a(32)-a(33) from Giovanni Resta, Jun 26 2017
STATUS
approved