Search: id:A063990
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%I A063990
%S A063990 220,284,1184,1210,2620,2924,5020,5564,6232,6368,10744,10856,12285,
%T A063990 14595,17296,18416,63020,66928,66992,67095,69615,71145,76084,79750,
%U A063990 87633,88730,100485,122265,122368,123152,124155,139815,141664,142310
%N A063990 Amicable numbers.
%C A063990 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
%C A063990 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]
%D A063990 Scott T. Cohen, Mathematical Buds, Ed. H. D. Ruderman, Vol. 1 Chap. VIII
pp. 103-126 Mu Alpha Theta 1984.
%D A063990 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers,
pp. 145-7, Penguin Books 1987.
%D A063990 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 90.
%H A063990 T. D. Noe, Amicable numbers less than 10^12, table
of n, a(n) for n=1..15109 (from Pedersen's tables)
%H A063990 T. Andreescu, Number Theory Trivia: Amicable Numbers
%H A063990 T. Andreescu,
Number Theory Trivia: Amicable Numbers
%H A063990 Anonymous,
Amicable Pairs Applet Test
%H A063990 Anonymous,
Amicable and Social Numbers
%H A063990 G. D'Abramo, On Amicable
Numbers With Different Parity
%H A063990 L. Euler, On amicable
numbers
%H A063990 M. Garcia, A Million New Amicable Pairs, J. Integer Sequences,
4 (2001), #01.2.6.
%H A063990 M. Garcia, J. M. Pedersen and H. J. J. te Riele, Amicable Pairs, a Survey
%H A063990 Hisanori Mishima, Amicable Numbers:first 236 pairs(smaller member<10^8)
fully factorized
%H A063990 D. Moews, A List
Of The First 5001 Amicable Pairs
%H A063990 D. and P. C. Moews, A List Of Amicable
Pairs Below 2.01*10^11
%H A063990 J. O. M. Pedersen, Known
Amicable Pairs
%H A063990 J. O. M. Pedersen, Tables
of Aliquot Cycles
%H A063990 I. Peterson, MathTrek,
Appealing Numbers
%H A063990 I. Peterson, MathTrek,
Amicable Pairs, Divisors and a New Record
%H A063990 H. J. J. te Riele, On Generating New Amicable Pairs from Given
Amicable Pairs
%H A063990 H. J. J. te Riele, Computation of All the Amicable Pairs Below
10^10
%H A063990 H. J. J. te Riele,
A New Method for Finding Amicable Pairs
%H A063990 E. Sandifer, Amicable numbers
%H A063990 G. Villemin's Almanach of Numbers, Nombres amiables et sociables
%H A063990 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics
%H A063990 Wikipedia, Amicable
number
%H A063990 Number Theory List,
NMBRTHRY Archives--August 1993
%t A063990 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
%Y A063990 Union of A002025 and A002046.
%Y A063990 Sequence in context: A157673 A064477 A121507 this_sequence A157107 A135807
A102073
%Y A063990 Adjacent sequences: A063987 A063988 A063989 this_sequence A063991 A063992
A063993
%K A063990 nonn
%O A063990 1,1
%A A063990 N. J. A. Sloane (njas(AT)research.att.com), Sep 18 2001
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