A348148 - OEIS

%I #6 Nov 01 2021 21:18:04

%S 3780,66960,167400,406224,1097280,6656832,13035330,29410290,

%T 4529295360,27477725184,88071903612,1159632322560,7035102756864,

%U 18554223329280,22385029489560,54934276752360,112562288197632,125356165141536,307631949813216

%N Numbers k for which sigma(k)/k = 32/9.

%C This sequence will contain terms of the form 135*P and 819*Q, where P is a perfect number (A000396) not divisible by 3 or 5, and Q is a perfect number not divisible by 3, 7, or 13. Proof: sigma(135*P)/(135*P) = sigma(135)*sigma(P)/(135*P) = 240*(2*P)/(135*P) = 32/9 and sigma(819*Q)/(819*Q) = sigma(819)*sigma(Q)/(819*Q) = 1456*(2*Q)/(819*P) = 32/9. QED

%C Terms ending in "4", "32", or "80" and some terms ending in "60" will have one of these forms:

%C   a( 1) =            3780 = 135*          28 = 135*A000396(2)

%C   a( 2) =           66960 = 135*         496 = 135*A000396(3)

%C   a( 4) =          406224 = 819*         496 = 819*A000396(3)

%C   a( 5) =         1097280 = 135*        8128 = 135*A000396(4)

%C   a( 6) =         6656832 = 819*        8128 = 819*A000396(4)

%C   a( 9) =      4529295360 = 135*    33550336 = 135*A000396(5)

%C   a(10) =     27477725184 = 819*    33550336 = 819*A000396(5)

%C   a(12) =   1159632322560 = 135*  8589869056 = 135*A000396(6)

%C   a(13) =   7035102756864 = 819*  8589869056 = 819*A000396(6)

%C   a(14) =  18554223329280 = 135*137438691328 = 135*A000396(7)

%C   a(17) = 112562288197632 = 819*137438691328 = 819*A000396(7).

%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiperfect">Multiperfect Numbers and Hemiperfect Numbers</a>

%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy: Some Resources (preliminary version 4)</a>

%H Walter Nissen, <a href="http://upforthecount.com/math/ffp8.html">Primitive Friendly Pairs with friends < 2^34 with denom < 20000</a>

%e 167400 is a term, since sigma(167400)/167400 = 595200/167400 = 32/9.

%t Select[Range[5*10^8], DivisorSigma[1, #]/# == 32/9 &]

%t Do[If[DivisorSigma[1, k]/k == 32/9, Print[k]], {k, 5*10^8}]

%Y Cf. A000203, A000396, A211680, A212610.

%Y Subsequence of A005101 and A218416.

%K nonn

%O 1,1

%A _Timothy L. Tiffin_, Oct 02 2021