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%I #15 Oct 26 2019 14:32:05
%S 24,30,40,42,48,60,80,80,96,102,126,140,140,156,156,156,174,180,180,
%T 198,216,224,224,264,276,280,294,294,300,320,340,372,380,384,440,440,
%U 468,500,504,510,528,560,582,608,616,642,680,684,690,690,696,702,736,750
%N First of a pair of compatible numbers, where two numbers m and n are compatible if sigma(n)-2dn=sigma(m)-2dm=m+n, for some proper divisors dn and dm of m and n respectively.
%C Compatible numbers were introduced by Sachs in analogy to amicable numbers, as admirable numbers are analogous to perfect numbers. Some terms have more than one counterpart (A109798), like 80 (two counterparts: 102 and 104) or 156 (3 counterparts: 210, 230 and 234). - _Amiram Eldar_, Oct 26 2019
%H Amiram Eldar, <a href="/A109797/b109797.txt">Table of n, a(n) for n = 1..1000</a>
%H J. M. Sachs, <a href="https://www.jstor.org/stable/41184328">Admirable Numbers and Compatible Pairs</a>, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
%H T. Trotter, <a href="http://www.trottermath.net/numthry/admirabl.html">Admirable Numbers.</a> [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - _N. J. A. Sloane_, Mar 29 2018]
%e sigma(42)-2(1)=96-2=94 and sigma(52)-2(2)=98-4=94 and 42+52=94 so a(4)=42.
%p L:=remove(proc(z) isprime(z) end, [$1..10000]): S:=proc(n) map(proc(z) sigma(n) -2*z end, divisors(n) minus {n}) end; CK:=map(proc(z) [z,S(z)] end, L): CL:=[]: for j from 1 to nops(CK)-1 do x:=CK[j,1]; sx:=sigma(x); Sx:=CK[j,2]; for k from j+1 to nops(CK) while CK[k,1]<sx do y:=CK[k,1]; if x+y in Sx intersect CK[k,2] then CL:=[op(CL),[x,y,x+y]] fi od od;
%t seq = {}; Do[d = Most[Divisors[n]]; s = Total[d]; Do[m = s - 2*d[[k]]; If[m <= 0 || m <= n, Continue[]]; delta = DivisorSigma[1, m] - m - n; If[delta > 0 && EvenQ[delta] && delta/2 < m && Divisible[m, delta/2], AppendTo[seq, n]], {k, Length[d], 1, -1}], {n, 1, 750}]; seq (* _Amiram Eldar_, Oct 26 2019 *)
%Y Cf. A109798, A111592.
%K nonn
%O 1,1
%A _Walter Kehowski_, Aug 15 2005
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