Problems & Puzzles:
Problems
Problem 9. Let f(n) be the number of ways of representing n (n can be a
composite or a prime number) as the sum of one or more consecutive primes. For
example :
f(0)=0, f(1)=0, f(2)=1, f(3)=1, f(4)=0, f(5)=2, f(6)=0, f(7)=1,
f(8)=1,…f(41)=3
5 : 5=2+3
41 : 41=2+3+5+7+11+13 =11+13+17
Leo Moser asks : Is f(n)=k solvable for every k?
I have gotten the least n for k=1,2,3,4,5
k 
least n 
Expression of the sum of consecutive
primes 
0 
9 (odd) 4
(even) 
N/A N/A 
1 
3 2 
3=3 2=2 
2 
5 36 
5=s(2à 3) 36=s(5à 13)=s(17à 19) 
3 
41 240 
41=s(2à 13)=s(11à 17)
240=s(17à
43)=s(53à 67)=s(113à 127) 
4 
1151 1164 
1151=(7à 101)=s(223à
239)=S(379à 389) 1164=s(29à 103)=s(97à 139)=s(281à 307)=s(577à 587) 
5 
311 863
14369
20272 
311=s(11à 47)=s(31à 59)=s(53à 71)=s(101à 107) 863=s(29à 89)=s(41à 97)=s(107à 139)=s(163à 181)
14369=s(53à
409)=s(173à 443)=s(491à 647) =s(7487à 4793)
20272=s(107à
499)=s(151à 509)=s(761à 929)=s(1217à 1307)=s(10133à
10139) 
6 
34421
130638 
34421=s(269à 709)=s(1429à
1571)=s(3793à 3853)=s(4889à 4937)=s(11467à 11483) The smallest even number for which f(n)
== 6 is 130638:
130638 == Sum of primes from 29 to 1319
130638 == Sum of primes from 461 to 1439
130638 == Sum of primes from 2113 to 2551
130638 == Sum of primes from 10847 to 10939
130638 == Sum of primes from 13009 to 13109
130638 == Sum of primes from 16273 to 16363 
7 
442019
218918 
The smallest odd number for which f(n) == 7 is 442019: 442019
== Sum of primes from 419 to 2621
442019 == Sum of primes from 7529 to 8017
442019 == Sum of primes from 13229 to 13567
442019 == Sum of primes from 17569 to 17807
442019 == Sum of primes from 49069 to 49157
442019 == Sum of primes from 147331 to 147347
442019 == Sum of primes from 442019 to 442019
218918 == Sum of primes from 3301 to 3769
218918 == Sum of primes from 4561 to 4957
218918 == Sum of primes from 5623 to 5897
218918 == Sum of primes from 7691 to 7937
218918 == Sum of primes from 9851 to 10069
218918 == Sum of primes from 13619 to 13729
218918 == Sum of primes from 18199 to 18289 
8 
3634531
9186778 
The smallest odd number for which f(n) == 8 is 3634531: 3634531
== Sum of primes from 313 to 7877
3634531 == Sum of primes from 977 to 7937
3634531 == Sum of primes from 31567 to 32713
3634531 == Sum of primes from 70997 to 71483
3634531 == Sum of primes from 73897 to 74419
3634531 == Sum of primes from 172969 to 173191
3634531 == Sum of primes from 519161 to 519257
3634531 == Sum of primes from 3634531 to 3634531
The smallest even number for which f(n) == 8 is 9186778:
Sum from 439 to 12853 is 9186778
Sum from 18433 to 22871 is 9186778
Sum from 52501 to 54371 is 9186778
Sum from 84443 to 85667 is 9186778
Sum from 176413 to 176951 is 9186778
Sum from 218513 to 218971 is 9186778
Sum from 353149 to 353501 is 9186778
Sum from 4593377 to 4593401 is 9186778 
9 
48205429 
The smallest number for which f(n) == 9 is 48205429: Sum
from 46507 to 56611 is 48205429
Sum from 124291 to 128749 is 48205429
Sum from 176303 to 179461 is 48205429
Sum from 331537 to 333397 is 48205429
Sum from 433577 to 434939 is 48205429
Sum from 541061 to 542149 is 48205429
Sum from 2536943 to 2537323 is 48205429
Sum from 16068461 to 16068499 is 48205429
Sum from 48205429 to 48205429 is 48205429 
10 
? 

The solutions for k=6, even, for k=7, 8 & 9 have been
obtained by Jack Brennen between 6&7 of September of 1998.
Find the least n values for k=>9 (even) and for k=10.
(Ref. 2, p. 107.108, C2)
