- Has anyone noticed that the recent Chermoni-Wroblewski 17-tuplet (message #22949) has this left to right gap pattern namely (4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4, 2) and that this set of 17 consecutive primes namely, (17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83) has a left to right gap pattern (2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4) that is the reverse of the 17-tuplet pattern?

Seventeen consecutive primes starting with 17 and with the same 17-tuplet gaps in exact reverse order, rather a neat coincidence what? Furthermore, the leftmost 3 digits of the initial term of the 17-tuplet "102" and the rightmost 2 digits "17" are both divisible by 17. Ranaan and Jarek outdid themselves with this one!

Using Tony Forbes list of permissible prime k-tuple patterns, I found that each k up to and including 23 has a pattern that is the reverse of a pattern of some other set of k consecutive primes. But none with a prime k had a corresponding reverse pattern set whose initial term equaled k.

Does this reverse pattern correspondence hold true for any admissible prime k-tuplet? i.e. for any k>2, there exists a permissible prime k-tuplet and an associated set S of k consecutive odd positive primes, whose left to right gap pattern is the reverse of the left to right gap pattern of the tuplet.

Thanks folks.

Bill Sindelar

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"djbroadhurst" <d.broadhurst@...> wrote:

> The asymptotic number of perfect numbers less than x was proven in

The really tough question is whether the sum of the

>

> MR0090600 (19,837d)

> Hornfeck, Bernhard; Wirsing, Eduard

> Über die Häufigkeit vollkommener Zahlen. (German)

> Math. Ann. 133 (1957), 431-438.

>

> to be less than x^eps for all eps>0.

> From this it trivially follows that the sum of the reciprocals

> of the perfect numbers is finite.

>

> I regard Carl's result in

>

> MR0618552 (82m:10012)

> Pomerance, Carl

> On the distribution of amicable numbers. II.

> J. Reine Angew. Math. 325 (1981), 183-188.

>

> as "stronger" since it proves the finiteness of

> of the sum of the reciprocals of the amicable numbers.

reciprocals of the sociable numbers is finite.

One might guess so, from the opinions in

http://www.math.dartmouth.edu/~carlp/sociable.pdf

but there seems little hope of a proof,

given that we do not know whether an integer as small

as 276 is sociable.

David