A218407 -id:A218407 - cp's OEIS frontend

A218406 Numbers k for which sigma(k)/k - 1/5 is an integer.

5, 420, 7440, 8190, 18600, 121920, 131040, 95472000, 102136320, 197308800, 433305600, 503255040, 71271827200, 91967616000, 128848035840, 190066406400, 287879454720, 354560976000, 799959888000, 2061580369920, 2208930048000, 100368143884800, 341546502528000

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(15) > 10^11. - Donovan Johnson, Oct 31 2012
a(20) > 10^12. - Giovanni Resta, Nov 04 2012
Note that there are no terms here with abundancy 11/5 (k=2). - Michel Marcus, Jun 26 2013

Cf. A218407, A218408, A218427.

a(8)-a(14) from Donovan Johnson, Oct 31 2012
a(15)-a(19) from Giovanni Resta, Nov 04 2012
More terms from Michel Marcus, Jun 26 2013

A218408 Numbers k for which sigma(k)/k - 3/5 is an integer.

Original entry on oeis.org

15, 90, 3360, 2618880, 5059200, 4873497600, 7381524480, 27990144000, 57846297600, 4609292688000, 4440104532864000, 106644933787392000, 164741543366400000, 454310107592140800, 1084892424295680000, 1160492902900531200, 10932800808939609600

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(10) > 10^11. - Donovan Johnson, Oct 31 2012

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

Cf. A218406, A218407, A218427.

a(6)-a(9) from Donovan Johnson, Oct 31 2012

More terms from Michel Marcus, Jun 26 2013

A218427 Numbers k for which sigma(k)/k - 4/5 is an integer.

Original entry on oeis.org

10, 60, 1170, 114660, 72602880, 668304000, 714954240, 1307124000, 1381161600, 2701389600, 8052817920, 10181689600, 643773312000, 1330464844800, 2015156183040, 3522876144480, 15462510336000, 23885971200000, 702577007193600, 908714417345280

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(13) > 10^11. - Donovan Johnson, Oct 31 2012

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

M. Drennen, T. Gunter, P. Johnson and J. Ward, The Loneliest Number
Jeffrey Ward, Does Ten Have a Friend?, arXiv:0806.1001 [math.NT], 2008.

Cf. A218406, A218407, A218408.

a(5)-a(12) from Donovan Johnson, Oct 31 2012
a(13) from Giovanni Resta, Nov 04 2012
More terms from Michel Marcus, Jun 26 2013

A326200 Lexicographically earliest sequence such that a(i) = a(j) => sigma(i)/i = sigma(j)/j for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

Antti Karttunen, Jun 13 2019

nonn

Restricted growth sequence transform of the abundancy index of n.
For all i, j:
  a(i) = a(j) <=> A094759(i) = A094759(j),
  a(i) = a(j) => A017665(i) = A017665(j),
  a(i) = a(j) => A017666(i) = A017666(j).

Antti Karttunen, Table of n, a(n) for n = 1..105664
Achim Flammenkamp, The Multiply Perfect Numbers Page

Cf. A000396 (positions of 6's), A005820 (positions of 119's).
Cf. A017665, A017666, A074902, A094759.
Cf. A218406, A218407, A218408, A218427, A326202.

up_to = 105664; \\ (In the same equivalence class as 78, 364 and 6448).
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v326200 = rgs_transform(vector(up_to,n,sigma(n)/n));
A326200(n) = v326200[n];

A347222 Numbers k for which sigma(k)/k = 12/5.

Original entry on oeis.org

30, 140, 2480, 6200, 40640, 167751680, 42949345280, 687193456640, 11529215040699760640, 13292279957849158723273463079769210880, 957809713041180536473966890421518190654986607740846080, 65820182292848241686198767302293614551117361591934715588918640640

Offset: 1

Timothy L. Tiffin, Aug 23 2021

nonn

This sequence will contain terms of the form 5*P, where P is a perfect number (A000396) not divisible by 5. Proof: sigma(5*P)/(5*P) = sigma(5)*sigma(P)/(5*P) = 6*(2*P)/(5*P) = 12/5. QED

Terms ending in "30", "40", or "80" have this form. Example: a(n) = 5*A000396(n) for n = 1, 2, 3 and a(n) = 5*A000396(n-1) for n = 5..12.

			6200 is a term, since sigma(6200)/6200 = 14880/6200 = 12/5.

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.

Subsequence of A005101 and A218407.

Select[Range[5*10^8], DivisorSigma[1, #]/# == 12/5 &]
Do[If[DivisorSigma[1, k]/k == 12/5, Print[k]], {k, 5*10^8}]

a(9)-a(10) from Michel Marcus, Aug 24 2021

a(11)-a(12) from David A. Corneth, Aug 24 2021

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A218406 Numbers k for which sigma(k)/k - 1/5 is an integer.

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A218408 Numbers k for which sigma(k)/k - 3/5 is an integer.

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A218427 Numbers k for which sigma(k)/k - 4/5 is an integer.

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A326200 Lexicographically earliest sequence such that a(i) = a(j) => sigma(i)/i = sigma(j)/j for all i, j.

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A347222 Numbers k for which sigma(k)/k = 12/5.

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