A348021 Numbers k for which sigma(k)/k = 832/225.

94500, 195300, 1674000, 27432000, 56692800, 325883250, 735257250, 113232384000, 234013593600, 28990808064000, 59914336665600, 463855583232000, 559625737239000, 958634872012800, 1373356918809000, 7782220152472338432000, 16083254981776166092800, 8972288971548182138209587578844217344000

Offset: 1

Author

Timothy L. Tiffin, Sep 24 2021

Keywords

• nonn

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).

Examples

			325883250 is a term, since sigma(325883250)/325883250 = 1205043840/325883250 = 832/225.
		

Links

• G. P. Michon, Multiperfect Numbers and Hemiperfect Numbers
• Walter Nissen, Abundancy: Some Resources (preliminary version 4)
• Walter Nissen, Primitive Friendly Pairs with friends < 2^34 with denom < 20000

Crossrefs

Cf. A000203, A000396, A211680, A212610.

Subsequence of A005101.

Programs

Mathematica

Select[Range[5*10^8], DivisorSigma[1, #]/# == 832/225 &]
Do[If[DivisorSigma[1, k]/k == 832/225, Print[k]], {k, 5*10^8}]

Extensions

More terms from Michel Marcus, Oct 03 2021