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Sums of Primes

Site for tabulation of sums of primes, prime squares, prime cubes and various interesting properties of them.  

This site is a work in progress.

Prompted by the mersenneforum puzzle, I created a routine within YAFU (to leverage its prime generation capabilities) to sum squares of primes and test them for multiplicity of power of 10.  Charles R. Greathouse found the first several terms using a PARI one-liner.  These were verified by my routine and extended to include sums of primes up to 30 trillion, as of 31 March 2010.

This sequence appears in the OEIS, reference A174862.


 n         Pmax Sum(P^2) from 2 to Pmax; a multiple of 10^n
 1 907 37464550
 2 977 46403000
 3 977 46403000
 4 36643 1607722550000
 5 1067749 29911170219400000
 6 17777197 114513817910535000000
 7 71622461 6901046932669070000000
 8 2389799983 214033722754167115900000000
 9 31252968359 427048835201398147085000000000
 10 49460594569 1660700090685185957320000000000
 11 1915014433303 83775363722237720731978600000000000
 12 4076200167673 786646994677132840800629000000000000

I've also looked at sums of primes.  This sequence will be submitted to OEIS soon.
 n         Pmax Sum(P) from 2 to Pmax; a multiple of 10^n
 1 5 10
 2 23 100
 3 35677 63731000 
 4 106853 515530000 
 5 632501  15570900000 
 6 31190879  29057028000000 
 7 58369153  98078160000000 
 8 707712517  12606879200000000 
 9 26219976521  14640561651000000000 
 10 87424229843  154819819890000000000 
 11 1642257355619  48827520934500000000000
 12 2962734127453  155590824512000000000000

And sums of primes cubes.  Searched up to 10 trillion as of 7 April 2010
 n         Pmax Sum(P^3) from 2 to Pmax; a multiple of 10^n
 1 5 160
 2 233 143309500
 3 8783 167992435025000
 4 24763 9495929161130000
 5 5828099 18803849605481106073200000
 6 9229931 114943299218925309364000000
 7 262707241 62239590622437034770047320000000
 8 7717488553 39389391603365585735745579849700000000
 9 34529828929 14800565770732540706707662233175000000000
 10 311995561321 90365528187658782254536155073531290000000000
 11 549120448879 848814744633978332442418792098769600000000000
 12 3377754734499110532010356822624092227361649102207021134000000000000

The attached spreadsheet contains the prime, prime square, and prime cube sums in increments of 1 billion integers.  These can be fed into my program for sequence continuation, which someday I may return to.  

On a X5570 Xeon CPU, finding primes in each range of 1e9 takes about 3 seconds, and summing the primes/squares/cubes in that range (~ 30 - 40 million primes, depending on the interval) takes a fraction of a second (0.1 - 0.3 seconds).

Ĉ
sums.csv
(3475k)
Benjamin Buhrow,
Apr 7, 2010, 7:06 AM
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