Problems & Puzzles: Puzzles

Puzzle 959. Integers equal to, both, the sum of... Patrick Capelle found  (when?) this interesting curio: The sum of the primes from 2 to 357198967 is equal to the prime number 3325997869054417, and this is also equal to the sum of the non-primes integers from 1 to 83961758. Q1. Is there another larger solution, prime or not? Q2. Are there smaller prime solutions?

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Contributions came from Jeff Heleen, Paul Cleary

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Jeff wrote on July 1, 2019:

Q2. Using numbers from 1 to 65500 there are 4 solutions, only the
first (and smallest) of which is prime:                                                               

 

composites from 1 to:	primes from 2 to:	sum
4	3	5
9	11	28
414	967	71208
423	983	74139

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Paul wrote on July 5, 2019

Q1. I couldn’t find any other solution prime or not up to the sum of all primes up to 71981475733 with a total of 105788505266258967675.  The corresponding integers not prime were to 14873902548 with a total of 105788505279133031090.

 

Q2. There are 9 smaller solutions where the totals are the same, however only 1 of them is prime, i.e. 5.  Here are the sums of those that are the same.  {5,28,71208,74139,9260197734,12374540078,7574780746329,11101148723618,102581905748236}

Sorry it was just the way my program was written, here are the figures as you prefer them

 

{composite, prime, sum}

 

{4,3,5}

{9,11,28}

{414,967,71208}

{423,983,74139}

{142511,482593,9260197734}

{164637,561199,12374540078}

{4031808,15592813,7574780746329}

{4878612,18991039,11101148723618}

{14793444,59735051,102581905748236}

 

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