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%I #30 Aug 02 2017 23:37:59
%S 284,11697,38,369,26,11,286,3135,58,17,25,39,428,23,23,69,94,8225,244,
%T 41,31,87,478,59,82,41,118,267,142,71,4064,95,47,53,47,69,142,59,478,
%U 89,406,119,574,83,166,71,718,123,1292,205,71,97,418,143,71,295,79,89
%N The greater of the lexicographically least pair (x, y) such that 0 < x < y and sigma(x) = sigma(y) = x + y - n.
%C In the first 1000 terms the most repeated number is 719 with 14 occurrences.
%C What can we say about records in this sequence? - _David A. Corneth_, May 10 2017
%H David A. Corneth, <a href="/A285892/b285892.txt">Table of n, a(n) for n = 0..9999</a>
%e a(3) = 369: sigma(369) = sigma(180) = 369 + 180 - 3 = 546;
%e a(4) = 26: sigma(26) = sigma(20) = 26 + 20 - 4 = 42;
%e a(5) = 11: sigma(11) = sigma(6) = 11 + 6 - 5 = 12.
%e From _David A. Corneth_, May 10 2017 (Start):
%e a(35) = 69: sigma(62) = sigma(69) = 62 + 69 - 35 = 96.
%e After creating a list of pairs (sigma(i), i) and sorting them with respect to sigma(i), we get {[1, 1], [3, 2], [4, 3], [6, 5], [7, 4], [8, 7], [12, 6], [12, 11], [13, 9], ...}. Skimming through this list we see that the first pair of numbers having the same value for sigma are 6 and 11. As sigma(y) = x + y - n, we have n = x + y - sigma(y), giving n = 6 + 11 - 12 = 5. We have found no value for a(5) yet, therefore, a(5) = 11. (End)
%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(a); break;
%p fi; od; od; end: P(10^9);
%t Do[m = 1; While[Set[k, Module[{k = n + Boole[n == 0]}, While[! Xor[DivisorSigma[1, m] == DivisorSigma[1, k] == m + k - n, k >= m], k++]; k]] >= m, m++]; Print@ m, {n, 0, 50}] (* _Michael De Vlieger_, Apr 28 2017 (note: due to size of a(1) program takes a few minutes to run but posts results as soon as they are calculated.) *)
%o (PARI) upto(n, {u=50000}) = {my(res = vector(n,i,-1), v=vecsort(vector(u,i,[sigma(i), i])), t=1, u=2); while(u<=#v, if(v[t][1]==v[u][1], i=v[t][2] + v[u][2] - v[t][1]; if(1<=i && i<=n && res[i] == -1,res[i] = v[u][2]); u++, t++;u=t+1)); concat(284, res)} \\ (u is an estimate of the maximum of terms a(n) up to n) _David A. Corneth_, May 10 2017
%Y Cf. A000203, A095702, A285889, A285890.
%K nonn,easy
%O 0,1
%A _Paolo P. Lava_, Apr 28 2017
%E a(35) corrected by _David A. Corneth_, May 10 2017
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