Problems & Puzzles: Puzzles

 Puzzle 1090 Numbers that are the sum of k, k+2, ... consecutive primes Note: I forgot to celebrate our 24th anniversary three weeks ago. Suddenly I realized that we were days ahead of 23 of May... So, again thank you all folks for your splendid contributions sending or solving puzzle for my pages.          Numbers that are sums of  2, 4, 6, 8, and 10 consecutive primes (CPs): 30984360, 39851304, 215470920, 349764576, 414750412. None of them is sum of 12 CPs. Q1: More terms?  Q2: Cases including 12, 14,..CPs?

During the week from 11-17 June 2022, contributions came from Giorgos Kalogeropoulos, Emmanuel Vantieghem and Gennady Gusev

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Giorgios wrote:

Q1:  Numbers that are sums of  2, 4, 6, 8, and 10 CPs:
945210480,1622295444,1790916480,2945549310,3148002210,3228258516,3234907356,4683908800,5100924528,5766012000,
8397768900,9733355850,10591703664,10617384708,13004600354,19001086590,19910148722,21248225724,21573231530,
22254672816,23375037294,25357481304,25694908254,25809624500,26656887828,27441707362,28745305466,28825499966,
29016831060,30789860004,31837620450,34805481708,36399964510,37571696160,38278824636,38347010710,38824570600,
39215783694,42582447288,43732888792,44193331014,46327483716,47003464422,47846412384,48990803416,52314177924,
52925155240,53773455138,54390158826,55373598600,56282843768,56327267394,57221442790,58754911080,59891863988,
64117934220,64312838092,66259162908,66373258032,67322711156,67464295522,72371982420,74778004962,76023896886,
76414479422,77173664472,79844933376,80975055304,83326427388,84348615180,85952588018,87566842062,88297078890

Q2: Numbers that are sums of  2, 4, 6, 8, 10 and 12 CPs:
1622295444,31837620450,67464295522

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Emmanuel wrote:

First of all, the numbers  30984360, 215470920  and  349764576 in your list are not a sum of two consecutive primes.

The sequence should be 39851304, 414750412.
It continues with  945210480 (sum of 2, 4, 6, 8, 10  but not  12  CPs).
The next is  1622295444  and is also the sum of  12  CPs.
I continued the search for numbers that are sum of  2k  CPs with  k = 1, 2, ..., 6  and I found :
31837620450, 67464295522, 233805285570, 370398556558, 960387691512, 978423084312.
None of them is the sum of  14  CPs.  The first such number is certainly > 1065846890418.
But I guess it will be much bigger.
I came to this by looking at the smallest number  m(n)  that is sum of  2, 4, ..., 2n  CPs :
m(1) = 8
m(2) = 36
m(3) = 8412
m(4) = 227304
m(5) = 39851304
m(6) = 1622295444,
so, the next one may have 14  digits.

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For k=10 the correct list for sum <4296000000

39851304

414750412

945210480

1622295444

1790916480

2945549310

3148002210

3228258516

3234907356

For k=12

1622295444

The next numbers in example in puzzle  are not correct: they are not the sum of consecutive primes

30984360= 15492167+159492193, but 15492173 is between them.

215470920: 107735431, (107735437), 107735489

349764576: 174882283, (174882287), 174882293

The primes in brackets are missed.

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