This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A260310 #32 Sep 26 2016 21:01:41
%S A260310 3,8,7,16,11,18,7,27,17,45,29,50,41,54,53,60,31,64,71,84,29,99,107,
%T A260310 132,61,147,41,153,131,162,53,207,157,220,113,225,179,228,239,240,131,
%U A260310 242,79,243,73,245,127,255,127,256,229,264,223,280,113,297,199,315,73,325,317,336,181,338,43,343,269,348
%N A260310 Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members.
%C A260310 Consider pairs (x,y) of numbers where sum(p|x) p + sum(q|y) q = x*sum(p|x) 1/p + y*sum(q|y) 1/q where p, q are primes and sum(p|x) p > sum(q|y) q.
%C A260310 Or, pairs of numbers x and y where A008472(x) + A008472(y) = A069359(x) + A069359(y) where A008472(x) > A008472(y).
%C A260310 A001222(a(2n -1)) = 1 and A001222(a(2n)) >= 3.
%C A260310 For the vast majority of the time, a(2n-1) is prime. There seems to be about 1 pair per decade.
%C A260310 Conjecture: a(2n) < a(2n+2) for all n>0, but there are many times (1/10.84) that a(2n) + 1 = a(2n+2).
%C A260310 Conjecture: if a(2n-1) is prime then a(2n) is composite and vice versa. And when a(2n-1) is composite, it is congruent to 0 (mod 6). - _Robert G. Wilson v_, Jul 22 2015
%C A260310 The first conjecture appears to be satisfied because if both x and y were prime then the sum of the A008472 were the sum of the two primes and the sum of the A069359 were two. - _R. J. Mathar_, Aug 03 2015
%H A260310 Robert G. Wilson v, <a href="/A260310/b260310.txt">Table of n, a(n) for n = 1..19812</a>
%e A260310 3 and 8 is first pair of this sequence because A008472(3) + A008472(8) = 3 + 2 = 5 is equal to A069359(3) + A069359(8) = 1 + 4 = 5.
%t A260310 f[n_] := f[n] = Block[{fi = FactorInteger[n][[All, 1]]}, {Plus @@ fi, n*Plus @@ (1/fi)}] /; n > 0; k =3; lst = {}; While[ k < 400, j = 2; While[ j < k, If[ f[k][[1]] + f[j][[1]] == f[k][[2]] + f[j][[2]] && f[k][[1]] != f[k][[2]], AppendTo[lst, {j,k}]]; j++]; k++]; lst // Flatten (* _Robert G. Wilson v_, Jul 22 2015 *)
%Y A260310 Cf. A001222, A008472, A069359.
%K A260310 nonn
%O A260310 1,1
%A A260310 _Juri-Stepan Gerasimov_, Jul 22 2015
%E A260310 Corrected and edited by _Robert G. Wilson v_, Jul 22 2015