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%I #41 Feb 17 2022 03:56:30
%S 1,2,5,41,1151,311,34421,218918,3634531,48205429,1798467197,
%T 12941709050,166400805323,6123584726269
%N Smallest positive integer that can be expressed as the sum of consecutive primes in exactly n ways.
%C a(10)-a(12) found by Wilfred Whiteside in 2007. - _Giovanni Resta_, May 07 2020
%D R. K. Guy, Unsolved Problems In Number Theory, C2.
%H Leo Moser, <a href="https://doi.org/10.4153/CMB-1963-013-1">Notes on number theory. III. On the sum of consecutive primes</a>, Canad. Math. Bull. 6 (1963), pp. 159-161.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_046.htm">Puzzle 46. Primes expressible as sum of consecutive primes in K ways</a>, The Prime Puzzles and Problems Connection.
%e 41 = 41 = 11+13+17 = 2+3+5+7+11+13, 41 is the smallest number expressible in 3 ways, so a(3)=41.
%e From _Robert G. Wilson v_, Feb 21 2011: (Start)
%e a(0) = 1 because 1 cannot be expressed as the sum of any set of consecutive primes,
%e a(1) = 2 because 2 is the first prime,
%e a(2) = 5 because 2+3 = 5,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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,
%e 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)
%e From _Giovanni Resta_, May 07 2020: (Start)
%e The runs of primes corresponding to a(10)-a(13), in the format first prime (run length), are:
%e a(10) = 1798467197 (1), 599489047 (3), 51384499 (35), 41824483 (43), 14862469 (121), 2233859 (803), 1652909 (1083), 742243 (2371), 280591 (5683), 118297 (10073);
%e a(11) = 6470854519 (2), 2156951369 (6), 431390039 (30), 323542441 (40), 71896949 (180), 56266367 (230), 5574659 (2314), 4481189 (2874), 3547639 (3620), 1487399 (8366), 993197 (12024);
%e 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);
%e 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)
%t 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 *)
%Y Cf. A054845, A067381.
%K nonn,hard,more
%O 0,2
%A _Jud McCranie_, May 25 2000
%E a(10)-a(11) from _Bert Dobbelaere_, Apr 14 2020
%E a(12)-a(13) from _Giovanni Resta_, May 07 2020
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