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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 (list; graph; refs; listen; history; text; internal format)
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.

MAPLE

with(numtheory): P:=proc(n) local k, p; if not isprime(n) then

if -1/n+add(add(1/op(1, p)^k, k=1..op(2, p)), p=ifactors(n)[2])=1 then n;

fi; fi; end: seq(P(i), i=1..10^5); # Paolo P. Lava, Mar 12 2018

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

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Last modified May 29 09:35 EDT 2020. Contains 334697 sequences. (Running on oeis4.)