A285892 The greater of the lexicographically least pair (x, y) such that 0 < x < y and sigma(x) = sigma(y) = x + y - n.

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

Author

Paolo P. Lava, Apr 28 2017

Keywords

• nonn
• easy

Comments

In the first 1000 terms the most repeated number is 719 with 14 occurrences.

What can we say about records in this sequence? - David A. Corneth, May 10 2017

Examples

			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)
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999

Crossrefs

Cf. A000203, A095702, A285889, A285890.

Programs

Maple

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);

Mathematica

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.) *)

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

Extensions

a(35) corrected by David A. Corneth, May 10 2017