1,1

Theorem: If the three numbers p=3*(2^(n-1))-1, q=3*(2^n)-1 and r=9*(2^(2n-1))-1 are all prime where n>=2, then p*q*(2^n) and r*(2^n) are amicable numbers. This 9th century theorem is due to Thabit ibn Kurrah (See for example, the History of Mathematics by David M. Burton, 6th ed., p. 510). - Mohammad K. Azarian (azarian(AT)evansville.edu), May 19 2008

Conjecture:Let p = prime number; if 2^n=[2*(p+1)/3]; b=2*p+1 (with b=prime); c=2*p^2+4*p+1 (with c=prime); then p*b*2^n and c*2^n are amicable numbers. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 15 2009]

Scott T. Cohen, Mathematical Buds, Ed. H. D. Ruderman, Vol. 1 Chap. VIII pp. 103-126 Mu Alpha Theta 1984.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 145-7, Penguin Books 1987.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.

T. D. Noe, Amicable numbers less than 10^12, table of n, a(n) for n=1..15109 (from Pedersen's tables)

T. Andreescu, Number Theory Trivia: Amicable Numbers

T. Andreescu, Number Theory Trivia: Amicable Numbers

Anonymous, Amicable Pairs Applet Test

Anonymous, Amicable and Social Numbers

G. D'Abramo, On Amicable Numbers With Different Parity

L. Euler, On amicable numbers

M. Garcia, A Million New Amicable Pairs, J. Integer Sequences, 4 (2001), #01.2.6.

M. Garcia, J. M. Pedersen and H. J. J. te Riele, Amicable Pairs, a Survey

Hisanori Mishima, Amicable Numbers:first 236 pairs(smaller member<10^8) fully factorized

D. Moews, A List Of The First 5001 Amicable Pairs

D. and P. C. Moews, A List Of Amicable Pairs Below 2.01*10^11

J. O. M. Pedersen, Known Amicable Pairs

J. O. M. Pedersen, Tables of Aliquot Cycles

I. Peterson, MathTrek, Appealing Numbers

I. Peterson, MathTrek, Amicable Pairs, Divisors and a New Record

H. J. J. te Riele, On Generating New Amicable Pairs from Given Amicable Pairs

H. J. J. te Riele, Computation of All the Amicable Pairs Below 10^10

H. J. J. te Riele, A New Method for Finding Amicable Pairs

E. Sandifer, Amicable numbers

G. Villemin's Almanach of Numbers, Nombres amiables et sociables

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

Wikipedia, Amicable number

Number Theory List, NMBRTHRY Archives--August 1993

s[n_] := DivisorSigma[1, n] - n; AmicableNumberQ[n_] := If[Nest[s, n, 2] == n && ! s[n] == n, True, False]; Select[Range[10^6], AmicableNumberQ[ # ] &] - Ant King (mathstutoring(AT)ntlworld.com), Jan 02 2007

Union of A002025 and A002046.

Sequence in context: A157673 A064477 A121507 this_sequence A157107 A135807 A102073

Adjacent sequences: A063987 A063988 A063989 this_sequence A063991 A063992 A063993

nonn

N. J. A. Sloane (njas(AT)research.att.com), Sep 18 2001