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%I #34 May 14 2017 22:30:26
%S 220,48,174,390,102,280,160,500,66,132,54,24280,992,560,140,168,60,
%T 10360,1120,1232,198,210,132,2170,520,1520,96,168,330,732,60,4424,270,
%U 540,144,1000,1484,4080,220,840,1144,16500,1988,5456,210,528,150,4158,1180,12236
%N The smaller of the lexicographically least pair (x, y) such that 0 < x < y and sigma(x) = sigma(y) = n + x + y.
%C In the first 1000 terms the most repeated number is 840 with 15 occurrences.
%H Paolo P. Lava, <a href="/A285889/b285889.txt">Table of n, a(n) for n = 0..1000</a>
%e a(0) = 220: sigma(220) = sigma(284) = 220 + 284 = 504;
%e a(1) = 48: sigma(48) = sigma(75) = 48 + 75 + 1 = 124;
%e a(2) = 174: sigma(174) = sigma(184) = 174 + 184 + 2 = 360.
%p with(numtheory): P:=proc(q) local a,b,k,n; for n from 0 to q do for k from 1 to q do
%p a:=sigma(k)-k-n; b:=sigma(a)-a-n; if a>0 and b=k and a<>b then print(k); break;
%p fi; od; od; end: P(10^9);
%t Table[m = 1; While[MissingQ@ Set[k, SelectFirst[Range[m - 1], DivisorSigma[1, m] == DivisorSigma[1, #] == m + # + n &]], m++]; {k, m}, {n, 0, 10}][[All, 1]] (* Version 10.2, or *)
%t Do[m = 1; While[Set[k, Module[{k = 1}, While[! Xor[DivisorSigma[1, m] == DivisorSigma[1, k] == m + k + n, k >= m], k++]; k]] >= m, m++]; Print@ k, {n, 0, 10}] (* _Michael De Vlieger_, Apr 28 2017 *)
%o (PARI) getfirstterms(n)={my(L:list,S:list,k:small,t);L=List();S=List([1,3]);k=0;forstep(i=3,+oo,1,listput(S,sigma(i));forvec(j=[[2,i],[2,i]],t=vecsum(j)+k;if((S[j[1]]==t)&&(t==S[j[2]]),listput(L,j[1]);if(k==n,break(2),k++)),2));return(Vec(L))} \\ _R. J. Cano_, May 03 2017
%Y Cf. A000203, A285890, A285892.
%Y See first terms of A002025 and A005276.
%K nonn,easy
%O 0,1
%A _Paolo P. Lava_, Apr 28 2017
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