A285890
The greater of the lexicographically least pair (x, y) such that 0 < x < y and sigma(x) = sigma(y) = n + x + y.
Original entry on oeis.org
284, 75, 184, 615, 110, 435, 212, 585, 70, 195, 56, 30429, 1012, 915, 182, 297, 92, 16983, 1886, 1725, 250, 345, 182, 2415, 716, 2175, 130, 285, 506, 975, 78, 5145, 418, 1107, 225, 1305, 1504, 9275, 246, 2001, 1336, 35875, 2002, 6405, 322, 915, 176, 7315, 1292
Offset: 0
a(3) = 615: sigma(615) = sigma(390) = 615 + 390 + 3 = 1008;
a(4) = 110: sigma(110) = sigma(102) = 110 + 102 + 4 = 216;
a(5) = 435: sigma(435) = sigma(280) = 435 + 280 + 5 = 720.
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with(numtheory): P:=proc(q) local a,b,k,n; for n from 0 to q do for k from 1 to q do
a:=sigma(k)-k-n; b:=sigma(a)-a-n; if a>0 and b=k and a<>b then print(a); break;
fi; od; od; end: P(10^9);
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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 *)
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@ m, {n, 0, 10}] (* Michael De Vlieger, Apr 28 2017 *)
A285892
The greater of the lexicographically least pair (x, y) such that 0 < x < y and sigma(x) = sigma(y) = x + y - n.
Original entry on oeis.org
284, 11697, 38, 369, 26, 11, 286, 3135, 58, 17, 25, 39, 428, 23, 23, 69, 94, 8225, 244, 41, 31, 87, 478, 59, 82, 41, 118, 267, 142, 71, 4064, 95, 47, 53, 47, 69, 142, 59, 478, 89, 406, 119, 574, 83, 166, 71, 718, 123, 1292, 205, 71, 97, 418, 143, 71, 295, 79, 89
Offset: 0
a(3) = 369: sigma(369) = sigma(180) = 369 + 180 - 3 = 546;
a(4) = 26: sigma(26) = sigma(20) = 26 + 20 - 4 = 42;
a(5) = 11: sigma(11) = sigma(6) = 11 + 6 - 5 = 12.
From _David A. Corneth_, May 10 2017 (Start):
a(35) = 69: sigma(62) = sigma(69) = 62 + 69 - 35 = 96.
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)
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with(numtheory): P:=proc(q) local a,b,k,n; for n from 0 to q do for k from 1 to q do
a:=sigma(k)-k+n; b:=sigma(a)-a+n; if a>0 and b=k and a<>b then print(a); break;
fi; od; od; end: P(10^9);
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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.) *)
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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
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