A326137 - OEIS

%I #20 Nov 05 2025 15:22:42

%S 17342325,22678425,31674825,38686725,41420925,45090045,49358925,

%T 51740325,54033525,54695025,67660425,68939325,70703325,75818925,

%U 76392225,77106645,78217425,81375525,92400525,96316605,97383825,98750925,99147825,102284325,107694405,113656725,115420725,117890325,118728225,120536325,127766925

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

%C 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.

%C 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.

%H Antti Karttunen, <a href="/A326137/b326137.txt">Table of n, a(n) for n = 1..1032; all terms < 2^31</a>

%H Charles Greathouse and Eric W. Weisstein, <a href="https://mathworld.wolfram.com/OddPerfectNumber.html">MathWorld: Odd perfect number</a>

%H Oliver Knill, <a href="http://www.math.harvard.edu/~knill/seminars/perfect/handout.pdf">The oldest open problem in mathematics</a>, Handout for NEU Math Circle, December 2, 2007

%H P. P. Nielsen, <a href="https://arxiv.org/abs/math/0602485">Odd Perfect Numbers Have At Least Nine Distinct Prime Factors</a>, arXiv:math/0602485 [math.NT], 2006.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers">Perfect number: Odd perfect numbers</a>

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%o (PARI)

%o 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));

%o isA326137(n) = ((omega(n)>=5)&&isA228058(n));

%Y Subsequence of A228058.

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

%K nonn

%O 1,1

%A _Antti Karttunen_, Jun 12 2019