A212610 -id:A212610 - cp's OEIS frontend

A211680 Pairs of deficient numbers having the same value of sigma(k)/k in the order that they are found.

135, 819, 3375, 6975, 1485, 9009, 2295, 13923, 2565, 15561, 3105, 18837, 3915, 23751, 4185, 25389, 4995, 30303, 5535, 33579, 5805, 35217, 6345, 38493, 25137, 40131, 7155, 43407, 7965, 48321, 8235, 49959, 9045, 54873, 9585, 58149, 9855, 59787, 10665, 64701

Offset: 1

T. D. Noe, May 23 2012

nonn

These are pairs of friendly deficient numbers. The terms a(2k) are in order, but terms a(2k-1) may be out of order.
Many of these pairs (a,b) have the property that (k*a,k*b) is another pair for some integer k. See A212610 for primitive pairs.
From Suyash Pandit, Oct 13 2023: (Start)
The first (but not smallest) even term of this sequence is n=1278316. It is friendly to m=1680705, with sigma(n)/n = sigma(m)/m = 336/169.
The first pair of even terms in this sequence is (n,m) = (366776,1581644246) with sigma(n)/n = sigma(m)/m = 720/361.
It is possible to have more than two deficient numbers with the same value of sigma(n)/n. For example, the numbers 119129783409, 217416788955, and 1318995186327 all satisfy sigma(n)/n = 3584/1891. (End)

T. D. Noe, Table of n, a(n) for n = 1..2860

Cf. A005100 (deficient numbers), A212608, A212609 (the pairs separated).

Cf. A212610 (primitive pairs).

nn = 10^5; t = DivisorSigma[1, Range[nn]]/Range[nn]; t2 = Transpose[Select[Tally[t], #[[1]] < 2 && #[[2]] > 1 &]][[1]]; Sort[Table[Flatten[Position[t, t2[[n]]]], {n, Length[t2]}], #1[[2]] < #2[[2]] &]

A212611 Primitive deficient numbers n having a companion m > n such that sigma(n)/n = sigma(m)/m (ordered by increasing (smallest possible) m).

Original entry on oeis.org

135, 3375, 25137, 68607, 22113, 557375, 451737, 1278316, 273, 6739509, 188325, 8775, 9020375, 20365625, 1083537, 3416875, 17875, 1995, 171387125, 382536063, 49166, 366776, 1487640245, 417725

Offset: 1

T. D. Noe, May 23 2012

These numbers n = a(k) are listed in the order of increasing companions m = A212612(k). All these numbers appear to have only one companion.

Cf. A211680 (pairs of deficient numbers).

Cf. A212610 (pairs of primitive deficient numbers).

a(n) = A212610(2n-1). - M. F. Hasler, Feb 22 2013

a(10)-a(18) from Donovan Johnson, May 24 2012

a(19)-a(24) from Donovan Johnson, Jun 06 2012

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

Original entry on oeis.org

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

Offset: 1

Timothy L. Tiffin, Sep 24 2021

nonn

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).

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

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.

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

More terms from Michel Marcus, Oct 03 2021

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}]

cp's OEIS Frontend

A211680 Pairs of deficient numbers having the same value of sigma(k)/k in the order that they are found.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A212611 Primitive deficient numbers n having a companion m > n such that sigma(n)/n = sigma(m)/m (ordered by increasing (smallest possible) m).

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica