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 A054859 Smallest positive integer that can be expressed as the sum of consecutive primes in exactly n ways. 16
 1, 2, 5, 41, 1151, 311, 34421, 218918, 3634531, 48205429, 1798467197, 12941709050, 166400805323, 6123584726269 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(10)-a(12) found by Wilfred Whiteside in 2007. - Giovanni Resta, May 07 2020 REFERENCES R. K. Guy, Unsolved Problems In Number Theory, C2. Leo Moser, Notes on number theory. III. On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), pp. 159-161. LINKS Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection. EXAMPLE 41 = 41 = 11+13+17 = 2+3+5+7+11+13, 41 is the smallest number expressible in 3 ways, so a(3)=41. From Robert G. Wilson v, Feb 21 2011: (Start) a(0) = 1 because 1 cannot be expressed as the sum of any set of consecutive primes, a(1) = 2 because 2 is the first prime, a(2) = 5 because 2+3 = 5, a(4) = 1151 because 7+11+13+17+19+23+29+31+37+41+43+47+53+59+61+67+71+73+79+83+89+97+101 = 223+227+229+233+239 = 379+383+389 = 1151, a(5) = 311 because 11+13+17+19+23+29+31+37+41+43+47 = 31+37+41+43+47+53+59 = 53+59+61+67+71 = 101+103+107 = 311, a(6) = 34421 because  269+271+...+701+709 = 1429+1433+...+1567+1571 = 3793+3797+3803+3821+3823+3833+3847+3851+3853 = 4889+4903+4909+4919+4931+4933+4937 = 11467+11471+11483 = 34421, a(7) = 218918 because 3301+3307+...+3767+3769 = 4561+4567+...+4951+4957 = 5623+5639+...+5881+5897 = 7691+7699+...+7933+7937 = 9851+9857+...+10067+10069  = 13619+13627+...+13723+13729 = 18199+18211+...+18287+18289, a(8) = 3634531 because 313+317+...+7873+7877 = 977+983+...+7933+7937 = 31567+31573+...+32707+32713 = 70997+70999+...+71479+71483 = 73897+73907+...+74413+74419 = 172969+172973+...+173189+173191 = 519161+519193+...+519247+519257 = 3634531, a(9) = 48205429 because 124291+124297+...+128747+128749 = 176303+176317+...+179453+179461 = 331537+331543+...+333383+333397 = 433577+433607+...+434933+434939 = 541061+541087+...+542141+542149 = 2536943+2536991+...+2537303+2537323 = 16068461+16068469+16068499 = 48205429, etc. (End) From Giovanni Resta, May 07 2020: (Start) The runs of primes corresponding to a(10)-a(13), in the format first prime (run length), are: a(10) = 1798467197 (1), 599489047 (3), 51384499 (35), 41824483 (43), 14862469 (121), 2233859 (803), 1652909 (1083), 742243 (2371), 280591 (5683), 118297 (10073); a(11) = 6470854519 (2), 2156951369 (6), 431390039 (30), 323542441 (40), 71896949 (180), 56266367 (230), 5574659 (2314), 4481189 (2874), 3547639 (3620), 1487399 (8366), 993197 (12024); a(12) = 166400805323 (1), 55466935091 (3), 18488978293 (9), 3025468583 (55), 155650259 (1069), 135604109 (1227), 50227297 (3311), 29640257 (5605), 19365569 (8561), 6284627 (25655), 3188819 (46977), 429467 (127483); a(13) = 6123584726269 (1), 360210866021 (17), 197534990813 (31), 124971116311 (49), 48217200953 (127), 40023427859 (153), 21188870723 (289), 13225879553 (463), 6166740911 (993), 3642804197 (1681), 2232410683 (2743), 992896649 (6167), 17062531 (311319). (End) MATHEMATICA lmt = 500000000; p = Prime@ Range@ PrimePi@ lmt; t = Table[0, {lmt}]; Do[s = 0; j = i; While[s = s + p[[j]]; s <= lmt, t[[s]]++; j++], {i, Length@ p}]; Table[ Position[t, n, 1, 1], {n, 0, 0}] (* Robert G. Wilson v, Feb 21 2011 *) CROSSREFS Cf. A054845, A067381. Sequence in context: A175172 A218057 A126469 * A076725 A059917 A255963 Adjacent sequences:  A054856 A054857 A054858 * A054860 A054861 A054862 KEYWORD nonn,hard,more AUTHOR Jud McCranie, May 25 2000 EXTENSIONS a(10)-a(11) from Bert Dobbelaere, Apr 14 2020 a(12)-a(13) from Giovanni Resta, May 07 2020 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)