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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

Table of n, a(n) for n=0..13.

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 October 29 17:37 EDT 2020. Contains 338067 sequences. (Running on oeis4.)