A218416 -id:A218416 - cp's OEIS frontend

A218414 Numbers k for which sigma(k)/k - 1/9 is an integer.

540, 3276, 58032, 950976, 1862190, 17660160, 3925389312, 1005014679552, 16080326885376, 17908023591648, 43947421401888, 92069057203200, 207726054681600, 411471933675264, 12363050673792000, 160893693946908502272, 269783631952374398976, 467406267507560908800

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(8) > 10^11. - Donovan Johnson, Nov 01 2012
a(8) > 10^12. - Giovanni Resta, Nov 04 2012
Note that there are no terms with abundancy 10/9 (k=1) or 19/9 (k=2). Michel Marcus, Jun 25 2013

Cf. A218430, A160320, A218415, A218416, A160321, A218417, A218418.

a(6)-a(7) from Donovan Johnson, Nov 01 2012

More terms from Michel Marcus, Jun 25 2013

A218415 Numbers k for which sigma(k)/k - 4/9 is an integer.

Original entry on oeis.org

9, 1782, 2160, 5400, 13104, 52141320, 117641160, 173365920, 6829038720, 12415092480, 13356796320, 104381747712, 513480135168, 1377031864320, 6578372828160, 26896578508800, 294208373809152, 1447285170659328, 3151812152130048, 7734746166732288

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(12) > 10^11. - Donovan Johnson, Nov 01 2012

a(14) > 10^12. - Giovanni Resta, Nov 04 2012

Cf. A218414, A218430, A160320, A218416, A160321, A218417, A218418.

a(6)-a(11) from Donovan Johnson, Nov 01 2012
a(12)-a(13) from Giovanni Resta, Nov 04 2012
More terms from Michel Marcus, Jun 25 2013

A218417 Numbers k for which sigma(k)/k - 7/9 is an integer.

Original entry on oeis.org

135, 216, 819, 2678400, 6780874383360, 15298997575680, 358160471832960, 878948428037760, 69640897897267200, 27548836016065625124864000, 114509071123027415138304000, 204540330952262537736192000, 32445066814696289084018688000, 42000317261222229165905510400

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(5) > 10^11. - Donovan Johnson, Nov 01 2012

a(5) > 10^12. - Giovanni Resta, Nov 04 2012

Cf. A218414, A218430, A160320, A218415, A218416, A160321, A218418.

More terms from Michel Marcus, Jun 26 2013

A218418 Numbers k for which sigma(k)/k - 8/9 is an integer.

Original entry on oeis.org

252, 4464, 73152, 7448760, 41713056, 48117888, 94112928, 301953024, 975576960, 1773584640, 10759746816, 46351678464, 77308821504, 103448378880, 196718837760, 233400061440, 409698051840, 939767546880

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(14) > 10^11. - Donovan Johnson, Nov 01 2012

a(19) > 10^12. - Giovanni Resta, Nov 04 2012

Cf. A218414, A218430, A160320, A218415, A218416, A160321, A218417.

a(5)-a(13) from Donovan Johnson, Nov 01 2012

a(14)-a(18) from Giovanni Resta, Nov 04 2012

A218430 Numbers k for which sigma(k)/k - 2/9 is an integer.

Original entry on oeis.org

54, 2744280, 6191640, 182494620, 653425920, 702989280, 27025270272, 72475361280, 76172903718912, 15224461545984768, 1688635722988634112, 5953066676035614584064, 1608903162935227680030720, 14600472124349965895417376, 2263986000385276007625523200

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(9) > 10^11. - Donovan Johnson, Nov 01 2012
a(9) > 10^12. - Giovanni Resta, Nov 04 2012
Note that there are no terms here with abundancy 11/9 (k=1) or 29/9 (k=3). - Michel Marcus, Jun 25 2013

Cf. A218414, A160320, A218415, A218416, A160321, A218417, A218418.

a(4)-a(8) from Donovan Johnson, Nov 01 2012

More terms from Michel Marcus, Jun 25 2013

A348148 Numbers k for which sigma(k)/k = 32/9.

Original entry on oeis.org

3780, 66960, 167400, 406224, 1097280, 6656832, 13035330, 29410290, 4529295360, 27477725184, 88071903612, 1159632322560, 7035102756864, 18554223329280, 22385029489560, 54934276752360, 112562288197632, 125356165141536, 307631949813216

Offset: 1

Timothy L. Tiffin, Oct 02 2021

nonn

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
Terms ending in "4", "32", or "80" and some terms ending in "60" will have one of these forms:
  a( 1) =            3780 = 135*          28 = 135*A000396(2)
  a( 2) =           66960 = 135*         496 = 135*A000396(3)
  a( 4) =          406224 = 819*         496 = 819*A000396(3)
  a( 5) =         1097280 = 135*        8128 = 135*A000396(4)
  a( 6) =         6656832 = 819*        8128 = 819*A000396(4)
  a( 9) =      4529295360 = 135*    33550336 = 135*A000396(5)
  a(10) =     27477725184 = 819*    33550336 = 819*A000396(5)
  a(12) =   1159632322560 = 135*  8589869056 = 135*A000396(6)
  a(13) =   7035102756864 = 819*  8589869056 = 819*A000396(6)
  a(14) =  18554223329280 = 135*137438691328 = 135*A000396(7)
  a(17) = 112562288197632 = 819*137438691328 = 819*A000396(7).

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

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

Cf. A000203, A000396, A211680, A212610.

Subsequence of A005101 and A218416.

Select[Range[5*10^8], DivisorSigma[1, #]/# == 32/9 &]
Do[If[DivisorSigma[1, k]/k == 32/9, Print[k]], {k, 5*10^8}]

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A218414 Numbers k for which sigma(k)/k - 1/9 is an integer.

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A218415 Numbers k for which sigma(k)/k - 4/9 is an integer.

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A218417 Numbers k for which sigma(k)/k - 7/9 is an integer.

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A218418 Numbers k for which sigma(k)/k - 8/9 is an integer.

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A218430 Numbers k for which sigma(k)/k - 2/9 is an integer.

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A348148 Numbers k for which sigma(k)/k = 32/9.

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Mathematica