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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(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.

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(2n-1) 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(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

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

Juri-Stepan Gerasimov, Jul 22 2015

EXTENSIONS

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

STATUS

approved

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Last modified July 12 16:42 EDT 2020. Contains 335665 sequences. (Running on oeis4.)