Problems & Puzzles: Puzzles

Puzzle 1234 Follow uo to Puzzle 1233


Paolo Lava sent the following nice puzzle:

A possible variant to Puzzle 1233 is to consider only prime numbers, namely: find the smallest prime that can be expressed as the sum of k different primes, simultaneously for all k’s between 2 and n, with n >= 2. 
It is clear that such a prime must be the greater prime in a twin prime pair (p, p+2) because the first condition is to be sum of 2 primes but this imply 2 + p, in other words one of the prime must be 2.

5 = 2 + 3;
19 = 2 + 17; 3 + 5 + 11; 
31 = 2 + 29; 3 + 11 + 17; 2 + 5 + 11 + 13;
43 = 2 + 41; 3 + 11 + 29; 2 + 5 + 7 + 29; 3 + 5 + 7 + 11 +  17;
61 = 2 + 59; 3 + 11 + 47; 2 + 5 + 7 + 47; 3 + 5 + 7 +17 + 29; 2 + 3 + 5 + 11 + 17 + 23;
103 = 2 + 101; 3 + 11 + 89; 2 + 3 + 19 + 79; 3 + 5 + 11 + 13 + 71; 2 + 13 + 17 + 19 + 23 + 29; 3 + 5 + 7 + 11 + 17 + 29 +  31;
103 = 2 + 101; 3 + 11 + 89; 2 + 3+ 19+79; 3 + 5+ 11 +  13 + 71; 2 + 13 + 17 + 19 + 23 + 29; 3 + 5 + 7 + 11 + 17 + 29 + 31; 2 + 3 + 5 + 7 + 11 + 13 + 19 + 43;

...  

But I've seen it already exist in OEIS:  A090700

A090700
a(n) is the smallest number which is simultaneously a prime, the sum of 2 distinct primes, the sum of 3 distinct primes, ... and the sum of n distinct primes.
0
2, 5, 19, 31, 43, 61, 103, 103, 139, 151, 199, 229, 283, 313, 421, 421, 523, 523, 643, 661, 811, 811, 1021, 1021, 1231, 1231, 1429, 1429, 1609, 1621, 1873, 1873, 2143, 2239, 2551, 2551, 2791, 2791, 3121, 3121, 3463, 3463, 3853, 3853, 4243, 4261, 4723

 

Thus, the only thing you could ask is to extend it after the 47th term (4723)


Q Can you verify and extend the sequence (after 4723) ?

From August 23-30, 2025 conributions came from Giorgos Kalogeropoulos, Oscar Volpatti, Emmanuel Vantieghem

 ***

Giorgos wrote:

Here are the first 200 terms
5, 19, 31, 43, 61, 103, 103, 139, 151, 199, 229, 283, 313, 421, 421, 523, 523, 643, 661, 811, 811, 1021, 1021, 1231, 1231, 1429, 1429, 1609, 1621, 1873, 1873, 2143, 2239, 2551, 2551, 2791, 2791, 3121, 3121, 3463, 3463, 3853, 3853, 4243, 4261, 4723, 4723, 5233, 5233, 5641, 5641, 6091, 6133, 6661, 6661, 7213, 7213, 7759, 7759, 8293, 8293, 8971, 8971, 9631, 9631, 10273, 10273, 10891, 10891, 11701, 11701, 12379, 12379, 13219, 13219, 13903, 13933, 14869, 14869, 15583, 15583, 16453, 16453, 17389, 17389, 18253, 18253, 19141, 19141, 20149, 20149, 21193, 21193, 22093, 22093, 23203, 23203, 24181, 24181, 25303, 25303, 26683, 26683, 27583, 27583, 28753, 28753, 30013, 30013, 31153, 31153, 32413, 32413, 33751, 33751, 34963, 34963, 36343, 36343, 37591, 37591, 39043, 39043, 40429, 40429, 41851, 41851, 43321, 43321, 44773, 44773, 46273, 46273, 47779, 47779, 49279, 49333, 50971, 50971, 52543, 52543, 54403, 54403, 55903, 55903, 57793, 57793, 59359, 59359, 61153, 61153, 62929, 62929, 64663, 64663, 66571, 66571, 68449, 68449, 70381, 70381, 72223, 72223, 74161, 74203, 76159, 76159, 78193, 78193, 80209, 80209, 82351, 82351, 84391, 84391, 86533, 86533, 88609, 88609, 90823, 90823, 93133, 93133, 95191, 95191, 97501, 97501, 99991, 99991, 102061, 102061, 104383, 104383, 106783, 106783, 109171, 109171, 111733, 111733, 114199

***

Oscar wrote:

I searched up to sums of 1000 distinct primes.
Largest solution found for puzzle 1234:
n = 1000, s = 3683413. 
See attached files:
Pu1234 OV a.txt for the first 1000 terms of OEIS sequence A090700;
Pu1234 OV b.txt for the minimal sums corresponding to solution 3683413.
I noticed that OEIS page about sequence A090700 has been updated, with David Corneth submitting the first 570 terms.
Curious timing, but I assume that he was unaware of your puzzle; all such terms agree with mine.

***

Emmanuel wrote:

After an intense attempt to correct the code I used in Puzzle 1233 I succeeded in finding an extension of the table in Puzzle 1234 ;
5,19,31,43,61,103,103,139,151,199,229,283,313,421,421,523,523,643,661,811,811,
1021,1021,1231,1231,1429,1429,1609,1621,1873,1873,2143,2239,2551,2551,2791,
2791,3121,3121,3463,3463,3853,3853,4243,4261,4723,4723,5233,5233,5641,5641,
6091,6133,6661,6661,7213,7213,7759,7759,8293,8293,8971,8971,9631,9631,10273,
10273,10891,10891,11701,11701,12379,12379,13219,13219,13903,13933,14869,14869,
15583,15583,16453,16453,17389,17389,18253,18253,19141,19141,20149,20149,21193,
21193,22093,22093,23203,23203,24181,24181,25303,25303,26683,26683,27583,27583,
28753,28753,30013,30013,31153,31153,32413,32413,33751,33751,34963,34963,36343,
36343,37591,37591,39043,39043,40429,40429,41851,41851,43321,43321,44773,44773,
46273,46273,47779,47779,49279,49333,50971,50971,52543,52543,54403,54403,55903,
55903,57793,57793,59359,59359,61153,61153,62929,62929,64663,64663,66571,66571,
68449,68449,70381,70381,72223,72223,74161,74203,76159,76159,78193,78193,80209,
80209,82351,82351,84391,84391,86533,86533,88609,88609,90823,90823,93133,93133,
95191,95191,97501,97501,99991,99991,102061,102061,104383,104383,106783,106783,
109171,109171,111733,111733}
Though my code let me refind the results of Giorgos Kalogeropoulos and Oscar Volpatti's in Puzzle 1233,.
I think that it is possible that some values are not minimal.

 

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