A066511 f-amicable numbers where f(n) = n-1.

100, 110, 1806, 1872, 2404, 3742, 12488, 14378, 25130, 26696, 53418, 57448, 61962, 64938, 67528, 67624, 172362, 187624, 195114, 208072, 591882, 643624, 790758, 938948, 1361562, 1381624, 1803776, 1877682, 1892224, 2091770, 3335288, 3559402, 6585656, 8810794

Offset

1,1

Comments

f-amicable pairs are defined similarly to f-perfect numbers in A066218. That is, a, b is a f-amicable pair if f(a) = D(b) and f(b) = D(a), where D(n) = sum_{k divides n, k<n} f(d).

Equivalently, let g(n) = sigma(n)-n-d(n)+2, where d(n) is the number of divisors of n and sigma(n) is their sum. Then n is in the sequence if g(g(n))=n but g(n) != n. (Sequence A066230 contains the solutions of g(n)=n.)

Links

Amiram Eldar, Table of n, a(n) for n = 1..162

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Example

Proper divisors of 100 = {1, 2, 4, 5, 10, 20, 25, 50}. f applied to these divisors = {0, 1, 3, 4, 9, 19, 24, 49}; their sum = 109. So D(100) = f(110). proper divisors of 110 = {1, 2, 5, 10, 11, 22, 55}. f applied to these divisors = {0, 1, 4, 9, 10, 21, 54}; their sum = 99. So D(110) = f(100). Therefore 100, 110 is an f-amicable pair.

Mathematica

g[ n_ ] := DivisorSigma[ 1, n ]-n-DivisorSigma[ 0, n ]+2; For[ n=1, True, n++, If[ g[ g[ n ] ]==n&&g[ n ]!=n, Print[ n ] ] ]

Crossrefs

Cf. A066230, A066218.

Sequence in context: A204590 A115454 A122466 * A281193 A285776 A226032

Adjacent sequences: A066508 A066509 A066510 * A066512 A066513 A066514

Keyword

nonn

Author

Joseph L. Pe, Jan 04 2002

Extensions

Edited by Dean Hickerson, Jan 10, 2002.

More terms from Amiram Eldar, Oct 02 2019

Status

approved