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A109797 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.

%I A109797 #17 Jun 05 2025 13:46:38
%S A109797 24,30,40,42,48,60,80,80,96,102,126,140,140,156,156,156,174,180,180,
%T A109797 198,216,224,224,264,276,280,294,294,300,320,340,372,380,384,440,440,
%U A109797 468,500,504,510,528,560,582,608,616,642,680,684,690,690,696,702,736,750
%N A109797 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 A109797 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 A109797 Amiram Eldar, <a href="/A109797/b109797.txt">Table of n, a(n) for n = 1..1000</a>
%H A109797 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 A109797 T. Trotter, <a href="http://www.trottermath.net/numthry/admirabl.html">Admirable Numbers.</a>
%e A109797 sigma(42)-2(1)=96-2=94 and sigma(52)-2(2)=98-4=94 and 42+52=94 so a(4)=42.
%p A109797 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 A109797 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 A109797 Cf. A109798, A111592.
%K A109797 nonn
%O A109797 1,1
%A A109797 _Walter Kehowski_, Aug 15 2005