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Recently Jonathan Sondow and Kieren MacMillan discovered that primary pseudoperfect numbers, when reduced modulo 288, form the arithmetic progression 6, 42, 78, 114, 150, 186, 222. They speculated that the ninth PPN, as yet undiscovered , will equal 258 , when reduced modulo 288.

This remarkable discovery prompted me to look at Giuga Numbers , reduced modulo 288, to see if a similar arithmetic progression exists in that case.
Only thirteen Giuga Numbers are known and they form the following sequence (which is not an arithmetic progression, but if you ignore the fact that some numbers are repeated several times, then it is an arithmetic progression !). The 13 known Giuga numbers , when reduced modulo 288, give us the following sequence: 30 , 282 , 282 , 246 , 210 , 210 , 174 , 174 , 174 , 138 , 138 , 138 , 66 .

A curious point is that the number of times a particular integer appears in this sequence (or progression ) seems to be related to the number of prime factors that a Giuga number possesses . For example , there is one Giuga Number with 3 prime factors , thus 30 appears only once in this arithmetic progression ; there are two Giuga Numbers with 4 prime factors (namely 858 and 1722 ) and thus 282 appears twice in this arithmetic progression; there is one Giuga Number with 5 prime factors (namely 66198 ) and thus 246 appears only once in this arithmetic progression; there are two Giuga Numbers with 6 prime factors , and thus 210 appears twice in this arithmetic progression ; Similarly, there are three Giuga Numbers with 7 prime factors , accounting for the three appearances of 174 in the pattern , and there are three Giuga Numbers with 8 prime factors , accounting for the three appearances of 138 in the pattern. The largest known Giuga Number (found by Frederick Schneider in 2006) which has 10 prime factors, when reduced modulo 288 , gives us a 66 in the arithmetic progression. Notice that 102 is missing in this sequence , presumably corresponding to the fact that no Giuga numbers with nine prime factors are known. Thus, there seems to be a relation between frequency of appearance in the sequence and the number of prime factors that a Giuga number possesses. Does anyone know why this should be the case ? Why should Giuga numbers, when reduced mod 288 , form this strange sequence, where frequency of appearance in the sequence seems to be related to the number of prime factors ?

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  • $\begingroup$ quadratic residues may be a help ? $\endgroup$ – user451844 Sep 16 '17 at 23:22
  • $\begingroup$ Is it only the frequency that is determined by the number of prime factors? It seems simpler to just conjecture that the residue is determined by the number of prime factors. This could be regarded as being related to your earlier question about Giuga numbers having exactly one prime factor of the form $4k+1$. $\endgroup$ – Erick Wong Jan 23 '18 at 0:54

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