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) Records   |  Conjectures  |  Problems  |  Puzzles  Home | Melancholia | Problems & Puzzles | References | News | Personal Page | Puzzlers | Search | Bulletin Board | Chat | Random Link Copyright © 1999-2012 primepuzzles.net. All rights reserved.