Sums of all Squares

[davar55]

2^2 + 3^2 + 5^2 + ... + p^2 = 10^mK

What is the smallest prime p such that
the sum of squares of all primes up to p
is a multiple of 10 (or 100 or 1000).

[axn]

s=0;forprime(p=2,1000,s=s+p^2;if(Mod(s,10)==0, print(p, ":",s)))
907:37464550
967:44505630
977:46403000
991:48351370

[davar55]

Nice and simple and quick reply -- thanks.

I won't ask about extending the list to 10000, etc. .....

[CRGreathouse]

907, 977, 977, 36643, 1067749, 17777197, 71622461, 2389799983, ...

The next term (if one exists) is more than 4 trillion.

[cheesehead]

Not yet in the OEIS.

http://www.research.att.com/~njas/sequences/

Search: 907, [B] 967, 977, 991[/B]

I am sorry, but the terms do not match anything in the table.

If your sequence is of general interest, please send it to me using the form provided and I will (probably) add it to the data base! Include a brief description and if possible enough terms to fill 3 lines on the screen. I need a minimum of 4 terms.

I think it qualifies. Also, I'm fond of OEIS entries with relatively large initial terms -- especially when the next few terms are so closely spaced as in this one. (Might it set some record in that regard -- highest ratio of initial term to average spacing of next n terms, for n = 3?)

I'd be glad to submit it, but I think it should be one of you guys.

How about generalizing to other bases?

[axn]

Originally Posted by cheesehead

Not yet in the OEIS.

I think CRG's sequence is more "worthy". It is also the solution of OP.

Originally Posted by cheesehead

How about generalizing to other bases?

I can think of two ways to generalize: to other bases and other powers (other than squares).

[cheesehead]

Originally Posted by axn

I think CRG's sequence is more "worthy".

Even with the omitted and repeated (just a typo) terms? :smile:

What I had in mind was a submission with the best of both your contributions.

I can think of two ways to generalize: to other bases and other powers (other than squares).

Oh, wow ... bases 2-16 or so, powers to, say, ninth ==> 135 sequences.

[axn]

Originally Posted by cheesehead

Even with the omitted and repeated terms?

That sequence is the first occurrence of 10^n. 977 repeats (not a typo!) because it ends in 000 and comes before any other 00. So it stands at positions 2 & 3.

EDIT:- Mine is merely the first four occurrences of 10

[cheesehead]

Originally Posted by axn

That sequence is the first occurrence of 10^n. 977 repeats because it ends in 000 and comes before any other 00. So it stands at positions 2 & 3.

(Sorry, CRG)

But that doubles the potential number of sequences. 270.

[CRGreathouse]

Originally Posted by CRGreathouse

The next term (if one exists) is more than 4 trillion.

That's *billion*, not trillion. Now my search limit is 50 billion, giving me

907, 977, 977, 36643, 1067749, 17777197, 71622461, 2389799983, 31252968359, 49460594569, ...

The nth term is very roughly n * log 10 * 10^n, so I was pretty lucky getting that last term. The next one will probably need over 2 trillion. Anyone up to the task? I don't actually have a good segmented sieve coded at the moment...

[bsquared]

Originally Posted by CRGreathouse

Anyone up to the task? I don't actually have a good segmented sieve coded at the moment...

I bit.

I have recently spent some time with my sieve, so decided to give this a shot.

I just started a run to 2 trillion. Here is the output so far:

found primes in range 0 to 1000000000 in elapsed time = 7.0227
**** 907 is 0 mod 10 ****
**** 977 is 0 mod 100 ****
**** 977 is 0 mod 1000 ****
**** 36643 is 0 mod 10000 ****
**** 1067749 is 0 mod 100000 ****
**** 17777197 is 0 mod 1000000 ****
**** 71622461 is 0 mod 10000000 ****
sum of squares complete in elapsed time = 8.5178, sum is 16352255694497179054764665

found primes in range 1000000000 to 2000000000 in elapsed time = 5.9418
sum of squares complete in elapsed time = 7.9423, sum is 126512354351558021982865866

found primes in range 2000000000 to 3000000000 in elapsed time = 5.9503
**** 2389799983 is 0 mod 100000000 ****
sum of squares complete in elapsed time = 7.7389, sum is 418923904898718760122282892

found primes in range 3000000000 to 4000000000 in elapsed time = 5.8990
sum of squares complete in elapsed time = 7.6150, sum is 979895993641271252685833855

found primes in range 4000000000 to 5000000000 in elapsed time = 5.8293
sum of squares complete in elapsed time = 7.4966, sum is 1894402266333772221759233898

With these timing trends, should have a result in 7 hours or so.

- ben.