cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A212610 Pairs of primitive deficient numbers having the same value of sigma(k)/k, listed by increasing value of the larger of the two k values.

Original entry on oeis.org

135, 819, 3375, 6975, 25137, 40131, 68607, 154791, 22113, 448335, 557375, 721525, 451737, 1401543, 1278316, 1680705, 273, 2876211, 6739509, 10590125, 188325, 11208375, 8775, 12525435, 9020375, 13934375, 20365625, 56162575, 1083537, 83479977, 3416875
Offset: 1

Views

Author

T. D. Noe, May 23 2012

Keywords

Comments

"Primitive" means that the pair (a(2n-1),a(n)) = (k,k') is not a multiple of another such pair occurring earlier. - M. F. Hasler, Feb 25 2013

Links

Crossrefs

Cf. A211680 (non-primitive pairs), A212611, A212612 (the pairs separated).

Formula

a(2n-1) = A212611(n), a(2n) = A212612(n), for all n>0. - M. F. Hasler, Feb 22 2013

Extensions

a(19)-a(31) from Donovan Johnson, May 24 2012

A212608 Deficient numbers n having a companion m > n such that sigma(n)/n = sigma(m)/m.

Original entry on oeis.org

135, 3375, 1485, 2295, 2565, 3105, 3915, 4185, 4995, 5535, 5805, 6345, 25137, 7155, 7965, 8235, 9045, 9585, 9855, 10665, 11205, 12015, 13095, 13635, 13905, 14445, 14715, 43875, 15255, 16335, 17145, 17685, 18495, 18765, 57375, 20115, 20385, 21195, 64125
Offset: 1

Views

Author

T. D. Noe, May 23 2012

Keywords

Comments

These numbers are listed in the order that their companions were found. All these numbers appear to have only one companion, which appear in A212609.

Crossrefs

Cf. A211680 (pairs of deficient numbers).

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

Views

Author

T. D. Noe, May 23 2012

Keywords

Comments

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.

Crossrefs

Cf. A211680 (pairs of deficient numbers).
Cf. A212610 (pairs of primitive deficient numbers).

Formula

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

Extensions

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

Views

Author

Timothy L. Tiffin, Sep 24 2021

Keywords

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

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

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

Views

Author

Timothy L. Tiffin, Oct 02 2021

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Cf. A000203, A000396, A211680, A212610.
Subsequence of A005101 and A218416.

Programs

  • Mathematica
    Select[Range[5*10^8], DivisorSigma[1, #]/# == 32/9 &]
Do[If[DivisorSigma[1, k]/k == 32/9, Print[k]], {k, 5*10^8}]
Showing 1-5 of 5 results.

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