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

Author

John Machacek, May 27 2017

Keywords

• nonn

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).

Examples

			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.
		

Links

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

Crossrefs

Cf. A000668, A007850.

Programs

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 *)

Extensions

a(20)-a(31) from Giovanni Resta, May 27 2017

a(32)-a(33) from Giovanni Resta, Jun 26 2017