

A260310


Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members.


4



3, 8, 7, 16, 11, 18, 7, 27, 17, 45, 29, 50, 41, 54, 53, 60, 31, 64, 71, 84, 29, 99, 107, 132, 61, 147, 41, 153, 131, 162, 53, 207, 157, 220, 113, 225, 179, 228, 239, 240, 131, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Consider pairs (x,y) of numbers where sum(px) p + sum(qy) q = x*sum(px) 1/p + y*sum(qy) 1/q where p, q are primes and sum(px) p > sum(qy) q.
Or, pairs of numbers x and y where A008472(x) + A008472(y) = A069359(x) + A069359(y) where A008472(x) > A008472(y).
A001222(a(2n 1)) = 1 and A001222(a(2n)) >= 3.
For the vast majority of the time, a(2n1) is prime. There seems to be about 1 pair per decade.
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).
Conjecture: if a(2n1) is prime then a(2n) is composite and vice versa. And when a(2n1) is composite, it is congruent to 0 (mod 6).  Robert G. Wilson v, Jul 22 2015
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


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..19812


EXAMPLE

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.


MATHEMATICA

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 *)


CROSSREFS

Cf. A001222, A008472, A069359.
Sequence in context: A101297 A224847 A309153 * A122237 A307162 A233418
Adjacent sequences: A260307 A260308 A260309 * A260311 A260312 A260313


KEYWORD

nonn


AUTHOR

JuriStepan Gerasimov, Jul 22 2015


EXTENSIONS

Corrected and edited by Robert G. Wilson v, Jul 22 2015


STATUS

approved



