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• f(1) and f(81) are the only square n s. Most common ten s: 0 , 5&6, 7, 3&8&9, 4, 2&1 Largest gap: between f(7) and f(30). 2,3,7,61,73,79,89 are the only prime
Message 1 of 3 , Feb 4, 2001
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f(1) and f(81) are the only square n's.

Most common ten's: 0 , 5&6, 7, 3&8&9, 4, 2&1

Largest gap: between f(7) and f(30).

2,3,7,61,73,79,89 are the only prime prime sum's of primes.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
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-----Original Message-----
From: Michael Bell [mailto:mdb36@...]
Sent: 04 February 2001 15:45
To: Primes List
Subject: Re: [PrimeNumbers] Sum of Primes

Hi,

I've just got Primeform out to give its recurrence mode a run on these. If
we define f(n) to be the sum of all primes <= p(2n) (where p(n) is the n'th
prime)) then:

f(1)
f(2)
f(3)
f(6)
f(7)
f(30)
f(32)
f(48)
f(50)
f(51)
f(54)
f(57)
f(61)
f(62)
f(65)
f(66)
f(73)
f(76)
f(79)
f(81)
f(89)
f(96)
f(99)
(1<=n<100)

are all prime, some randomly selected larger ones:
f(5001)
f(5010)
f(5011)
f(5012)
f(5022)
f(5029)
f(5031)
f(5084)
f(5086)
f(40025)
f(40032)
f(40035)
f(40039)
f(40052)
f(40060)
f(40065)
f(40070)
f(40071)
f(40082)
f(40091)
f(40099)
f(40100)
(these are PRP, could easily be proved by trial factoring if necessary)

Looking at bits of the list, it appears there is a great deal of clumping of
primes, and quite large stretches where 3 is a factor of f(n). Is there a
deep and meaningful reason for this?

Michael.

> Has anyone got any fascinating stories/URL's about prime sum's of primes.
>
> e.g.
>
> 2+3 is prime (5)
> 2+3+5+7 is prime (17)
> 2+3+5+7+11+13 is prime (41)
> 2+3+5+7+11+13+17+19 is not (77)
>
> etc...

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