## Wednesday, January 5, 2011

### Giuga numbers

Giuga numbers : numbers n such that p divides n/p - 1 for every prime divisor p of n.

30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506, ...

30 = 2*3*5 ..................... 1 = 1/2 + 1/3 + 1/5 - 1/30

858 = 2*3*11*13 ............ 1 = 1/2 + 1/3 + 1/11 + 1/13 - 1/858

1722 = 2*3*7*41 ............ 1 = 1/2 + 1/3 + 1/7 + 1/41 - 1/1722

66198 = 2*3*11*17*59 .... 1 = 1/2 + 1/3 + 1/11 + 1/17 + 1/59 - 1/66198

etc.

So far there are only 12 known Giuga numbers
[ n: p|(n/p-1) for every prime divisor p of n.]

The smallest one (= 30) has 3 prime factors (2, 3, 5)

4 factors:
858 = 2*3*11*13
1722 = 2*3*7*41

5 factors:
66198 = 2*3*11*17*59

6 factors:
2214408306 = 2*3*11*23*31*47057
24423128562 = 2*3*7*43*3041*4447

7 factors:
432749205173838 = 2*3*7*59*163*1381*775807
14737133470010574 = 2*3*7*71*103*67213*713863
550843391309130318 = 2*3*7*71*103*61559*29133437

8 factors:
244197000982499715087866346 =
2*3*11*23*31*47137*28282147*3892535183

554079914617070801288578559178 =
2*3*11*23*31*47059*2259696349*110725121051

1910667181420507984555759916338506 has 8 prime factors
(2 * 3 * 7 * 43 * 1831 * 138683 * 2861051 * 1456230512169437)

Could you find the next Giuga number?

Read first: Giuga's conjecture on primality

UPDATE!!

The number:
n=4200017949707747062038711509670656632404195753751630609228764416142557211582098432545190323474818

has 10 primes factors, but was not added to The On-Line Encyclopedia of Integer Sequences

4200017949707747062038711509670656632404195753751630609228764416142557211582098432545190323474818=
2*3*11*23*31*47059*2217342227*1729101023519*8491659218261819498490029296021*58254480569119734123541298976556403