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A326137
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Numbers with at least five distinct prime factors that satisfy Euler's criterion (A228058) for odd perfect numbers.
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5
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17342325, 22678425, 31674825, 38686725, 41420925, 45090045, 49358925, 51740325, 54033525, 54695025, 67660425, 68939325, 70703325, 75818925, 76392225, 77106645, 78217425, 81375525, 92400525, 96316605, 97383825, 98750925, 99147825, 102284325, 107694405, 113656725, 115420725, 117890325, 118728225, 120536325, 127766925
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OFFSET
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1,1
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COMMENTS
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P. P. Nielsen's 2006 paper shows that any odd perfect number must have at least nine distinct prime factors, thus if such numbers exist at all, they must occur in this sequence.
I conjecture that it is eventually possible to find an easy proof that this sequence has no common terms with A325981, and/or several other sequences (A326064, A326074, A326141, A326148, etc.) listed under index entry "sequences where odd perfect numbers must occur", thus settling the question about the existence of such numbers.
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LINKS
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PROG
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(PARI)
isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA326137(n) = ((omega(n)>=5)&&isA228058(n));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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