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A109798 Second 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 A109798 #15 Jun 05 2025 13:46:51
%S A109798 28,40,42,52,60,96,102,104,124,110,182,182,188,210,230,234,184,358,
%T A109798 362,204,312,248,252,408,372,424,306,388,418,434,376,516,384,508,530,
%U A109798 638,782,572,888,782,828,872,592,644,820,650,938,908,1026,1034,1102,976,760
%N A109798 Second 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 A109798 The terms are arranged by the order of their lesser counterparts (A109797). - _Amiram Eldar_, Oct 26 2019
%H A109798 Amiram Eldar, <a href="/A109798/b109798.txt">Table of n, a(n) for n = 1..1000</a>
%H A109798 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 A109798 T. Trotter, <a href="http://www.trottermath.net/numthry/admirabl.html">Admirable Numbers.</a>
%e A109798 sigma(42)-2(1)=96-2=94 and sigma(52)-2(2)=98-4=94 and 42+52=94 so a(4)=52.
%p A109798 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 A109798 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, m]], {k, Length[d], 1, -1}], {n, 1, 750}]; seq (* _Amiram Eldar_, Oct 26 2019 *)
%Y A109798 Cf. A109797, A111592.
%K A109798 nonn
%O A109798 1,1
%A A109798 _Walter Kehowski_, Aug 15 2005