A326137 Numbers with at least five distinct prime factors that satisfy Euler's criterion (A228058) for odd perfect numbers.

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

Offset: 1

Author

Antti Karttunen, Jun 12 2019

Keywords

• nonn

Comments

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1032; all terms < 2^31
• Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
• Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
• P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
• Wikipedia, Perfect number: Odd perfect numbers
• Index entries for sequences where any odd perfect numbers must occur

Crossrefs

Subsequence of A228058.

Cf. A001221, A325981, A326064, A326074, A326141, A326148.

Programs

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));