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.

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

Table of n, a(n) for n=1..33.

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

Cf. A000668, A007850.

Sequence in context: A080385 A120090 A280344 * A086830 A084699 A323441

Adjacent sequences: A286494 A286495 A286496 * A286498 A286499 A286500

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