OFFSET
1,1
COMMENTS
This sequence contains terms of the form 3375*P and 6975*Q, where P is a perfect number (A000396) not divisible by 3 or 5, and Q is a perfect number not divisible by 3, 5, or 31. Proof: sigma(3375*P)/(3375*P) = sigma(3375)*sigma(P)/(3375*P) = 6240*(2*P)/(3375*P) = 832/225 and sigma(6975*Q)/(6975*Q) = sigma(6975)*sigma(Q)/(6975*Q) = 12896*(2*Q)/(6975*P) = 832/225. QED
Many terms ending in "00" will have one of these forms:
a( 1) = 94500 = 3375* 28 = 3375*A000396(2)
a( 2) = 195300 = 6975* 28 = 6975*A000396(2)
a( 3) = 1674000 = 3375* 496 = 3375*A000396(3)
a( 4) = 27432000 = 3375* 8128 = 3375*A000396(4)
a( 5) = 56692800 = 6975* 8128 = 6975*A000396(4)
a( 8) = 113232384000 = 3375* 33550336 = 3375*A000396(5)
a( 9) = 234013593600 = 6975* 33550336 = 6975*A000396(5)
a(10) = 28990808064000 = 3375* 8589869056 = 3375*A000396(6)
a(11) = 59914336665600 = 6975* 8589869056 = 6975*A000396(6)
a(12) = 463855583232000 = 3375* 137438691328 = 3375*A000396(7)
a(14) = 958634872012800 = 6975* 137438691328 = 6975*A000396(7)
a(16) = 7782220152472338432000 = 3375*2305843008139952128 = 3375*A000396(8)
a(17) = 16083254981776166092800 = 6975*2305843008139952128 = 6975*A000396(8).
LINKS
G. P. Michon, Multiperfect Numbers and Hemiperfect Numbers
Walter Nissen, Abundancy: Some Resources (preliminary version 4)
EXAMPLE
325883250 is a term, since sigma(325883250)/325883250 = 1205043840/325883250 = 832/225.
MATHEMATICA
Select[Range[5*10^8], DivisorSigma[1, #]/# == 832/225 &]
Do[If[DivisorSigma[1, k]/k == 832/225, Print[k]], {k, 5*10^8}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Timothy L. Tiffin, Sep 24 2021
EXTENSIONS
More terms from Michel Marcus, Oct 03 2021
STATUS
approved