A054845 -id:A054845 - cp's OEIS frontend

A050936 Sum of two or more consecutive prime numbers.

5, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 36, 39, 41, 42, 48, 49, 52, 53, 56, 58, 59, 60, 67, 68, 71, 72, 75, 77, 78, 83, 84, 88, 90, 95, 97, 98, 100, 101, 102, 109, 112, 119, 120, 121, 124, 127, 128, 129, 131, 132, 138, 139, 143, 144, 150, 152, 155, 156, 158, 159, 160, 161, 162

Offset: 1

G. L. Honaker, Jr., Dec 31 1999

			E.g., 5 = (2 + 3) or (#2,2).
2+3 = 5, 3+5 = 8, 2+3+5 = 10, 7+5 = 12, 3+5+7 = 15, etc.

T. D. Noe, Table of n, a(n) for n = 1..10000
Patrick De Geest, WONplate 122
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Prime Sums

Subsequence of A034707.
Cf. A067372 up to A067381, A054996, A000040.
A084143(a(n)) > 0, complement of A087072.
Cf. A054845, A097889.

import Data.Set (empty, findMin, deleteMin, insert)
import qualified Data.Set as Set (null)
a050936 n = a050936_list !! (n-1)
a050936_list = f empty [2] 2 $ tail a000040_list where
   f s bs c (p:ps)
     | Set.null s || head bs <= m = f (foldl (flip insert) s bs') bs' p ps
     | otherwise                  = m : f (deleteMin s) bs c (p:ps)
     where m = findMin s
           bs' = map (+ p) (c : bs)
-- Reinhard Zumkeller, Aug 26 2011

# uses code of A084143
isA050936 := proc(n::integer)
    if A084143(n) >= 1 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 300 do
    if isA050936(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Aug 19 2020

lst={};Do[p=Prime[n];Do[p=p+Prime[k];AppendTo[lst, p], {k, n+1, 2*10^2}], {n, 2*10^2}];Take[Union[lst], 10^2] (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
f[n_] := Block[{len = PrimePi@ n}, p = Prime@ Range@ len; Count[ Flatten[ Table[ p[[i ;; j]], {i, len}, {j, i+1, len}],1], q_ /; Total@ q == n]]; Select[ Range@ 150, f@ # > 0 &] (* Or quicker for a larger range *)
lmt = 150; p = Prime@ Range@ PrimePi@ lmt; t = Table[0, {lmt}]; Do[s = 0; j = i+1; While[s = s + p[[j]]; s <= lmt, t[[s]]++; j++], {i, Length@ p}]; Select[ Range@ lmt, t[[#]] > 0 &] (* Robert G. Wilson v *)
Module[{nn=70,prs},prs=Prime[Range[nn]];Take[Union[Flatten[Table[Total/@ Partition[prs,i,1],{i,2,nn}]]],nn]] (* Harvey P. Dale, Nov 13 2013 *)

is(n)=my(v,m=1,t); while(1,v=vector(m++); v[m\2]=precprime(n\m); for(i=m\2+1,m, v[i]=nextprime(v[i-1]+1)); forstep(i=m\2-1,1,-1, v[i]=precprime(v[i+1]-1)); if(v[1]==0, return(0)); t=vecsum(v); if(t==n,return(1)); if(t>n, while(t>n,t-=v[m]; v=concat(precprime(v[1]-1), v[1..m-1]); t+=v[1]), while(tCharles R Greathouse IV, May 05 2016

list(lim)=my(v=List(),s,n=1,p); while(1, p=2; s=vecsum(primes(n++)); if(s>lim,return(Set(v))); listput(v,s); forprime(q=prime(n+1),, s+=q-p; if(s>lim,break); listput(v,s); p=nextprime(p+1))); \\ Charles R Greathouse IV, Nov 24 2021

More terms from David W. Wilson, Jan 13 2000

A054859 Smallest positive integer that can be expressed as the sum of consecutive primes in exactly n ways.

Original entry on oeis.org

1, 2, 5, 41, 1151, 311, 34421, 218918, 3634531, 48205429, 1798467197, 12941709050, 166400805323, 6123584726269

Offset: 0

Jud McCranie, May 25 2000

a(10)-a(12) found by Wilfred Whiteside in 2007. - Giovanni Resta, May 07 2020

			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)

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.
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A067381.

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 *)

a(10)-a(11) from Bert Dobbelaere, Apr 14 2020

a(12)-a(13) from Giovanni Resta, May 07 2020

A034707 Numbers that are sums (of a nonempty sequence) of consecutive primes.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 23, 24, 26, 28, 29, 30, 31, 36, 37, 39, 41, 42, 43, 47, 48, 49, 52, 53, 56, 58, 59, 60, 61, 67, 68, 71, 72, 73, 75, 77, 78, 79, 83, 84, 88, 89, 90, 95, 97, 98, 100, 101, 102, 103, 107, 109, 112, 113, 119, 120, 121, 124, 127

Offset: 1

Erich Friedman

A050936 is a subsequence (which still includes primes, embodied by A067377). - Enoch Haga, Jun 16 2002, R. J. Mathar, Oct 10 2010

T. D. Noe, Table of n, a(n) for n = 1..10000
Leo Moser, On the Sum of Consecutive Primes. Canad. Math. Bull. 6 (1963), 159-161.
Janyarak Tongsomporn, Saeree Wananiyaku, and Jörn Steuding, Sums of consecutive prime squares, Integers (2022) Vol. 22, #A9.

Complement is A050940.

Cf. A050936, A054845.

f[n_] := Block[{len = PrimePi@ n}, p = Prime@ Range@ len; Count[ Flatten[ Table[ p[[i ;; j]], {i, len}, {j, i, len}],1], q_ /; Total@ q == n]]; Select[ Range@ 1000, f@ # > 0 &] (* Or quicker for a larger range *)
lmt = 10000; 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}]; Select[ Range@ lmt, t[[#]] > 0 &]
upto=200;Select[Union[Flatten[Table[ Total/@Partition[Prime[ Range[ PrimePi[ upto]]],n,1],{n,upto-1}]]],#<=upto&] (* Harvey P. Dale, Jul 15 2011 *)

is(n)=if(isprime(n), return(1)); my(v,m=1,t); while(1, v=vector(m++); v[m\2]=precprime(n\m); for(i=m\2+1,m,v[i]=nextprime(v[i-1]+1)); forstep(i=m\2-1,1,-1,v[i]=precprime(v[i+1]-1)); if(v[1]==0, return(0)); t=vecsum(v); if (t==n, return(1)); if(t>n, while(t>n, t-=v[m]; v=concat(precprime(v[1]-1), v[1..m-1]); t+=v[1]), while(tCharles R Greathouse IV, May 05 2016

A054845(a(n)) > 0. - Ray Chandler, Sep 20 2023

Updated a misleading comment. - R. J. Mathar, Oct 10 2010

A054996 Integers that can be expressed as the sum of consecutive primes in exactly 1 way.

Original entry on oeis.org

2, 3, 7, 8, 10, 11, 12, 13, 15, 18, 19, 24, 26, 28, 29, 30, 37, 39, 42, 43, 47, 48, 49, 52, 56, 58, 61, 68, 73, 75, 77, 78, 79, 84, 88, 89, 95, 98, 102, 103, 107, 113, 121, 124, 128, 129, 132, 137, 144, 149, 150, 151, 155, 156, 157, 158, 159, 160, 161, 162, 163

Offset: 1

Jud McCranie, May 30 2000

nonn

			8=3+5, so 8 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054997, A054998, A054999, A055000, A055001.

A054845(a(n)) = 1. - Ray Chandler, Sep 20 2023

A054998 Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

Original entry on oeis.org

41, 83, 197, 199, 223, 240, 251, 281, 287, 340, 371, 401, 439, 491, 510, 593, 660, 733, 803, 857, 864, 883, 931, 941, 961, 983, 990, 991, 1012, 1060, 1061, 1099, 1104, 1187, 1236, 1283, 1313, 1361, 1381, 1392, 1433, 1439, 1493, 1511, 1523, 1524, 1553

Offset: 1

Jud McCranie, May 30 2000

nonn

			41 can be expressed as 41 or 11+13+17 or 2+3+5+7+11+13, so 41 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054999, A055500, A055001.

N:= 10^4: # to get all terms <= N
P:= [0,op(select(isprime, [2,seq(i,i=3..N,2)]))]:
nP:= nops(P);
S:= ListTools:-PartialSums(P):
V:= Vector(N):
for i from 1 to nP-1 do
  for j from i+1 to nP while S[j] - S[i] <= N do
     V[S[j]-S[i]]:= V[S[j]-S[i]]+1
od od:
select(t -> V[t] = 3, [$1..N]): # Robert Israel, Apr 05 2017

Module[{nn = 300, s}, s = Array[Prime, nn]; Keys@ Take[Select[KeySort@ Merge[Table[PositionIndex@ Map[Total, Partition[s, k, 1]], {k, nn/2}], Identity], Length@ # == 3 &], Floor[nn/6]]] (* Michael De Vlieger, Apr 06 2017, Version 10 *)

A054845(a(n)) = 3. - Ray Chandler, Sep 20 2023

A054999 Integers that can be expressed as the sum of consecutive primes in exactly 4 ways.

Original entry on oeis.org

1151, 1164, 1320, 1367, 1650, 1854, 1951, 2393, 2647, 2689, 2856, 2867, 3198, 3264, 3389, 3754, 4200, 4920, 4957, 5059, 5100, 5153, 5770, 5999, 6504, 7451, 7901, 8152, 8819, 10134, 10320, 10499, 10536, 10649, 10859, 10949, 11058, 12294

Offset: 1

Jud McCranie, May 30 2000

nonn

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054998, A055500, A055001.

A054845(a(n)) = 4. - Ray Chandler, Sep 20 2023

A054997 Integers that can be expressed as the sum of consecutive primes in exactly 2 ways.

Original entry on oeis.org

5, 17, 23, 31, 36, 53, 59, 60, 67, 71, 72, 90, 97, 100, 101, 109, 112, 119, 120, 127, 131, 138, 139, 143, 152, 173, 180, 181, 187, 204, 210, 211, 221, 228, 233, 258, 263, 269, 271, 276, 300, 304, 323, 330, 331, 349, 353, 372, 373, 379, 384, 390, 395, 408

Offset: 1

Jud McCranie, May 30 2000

nonn

			5 can be expressed as 5 or 2+3, so 5 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054998, A054999, A055500, A055001.

A054845(a(n)) = 2. - Ray Chandler, Sep 20 2023

A055001 Integers that can be expressed as the sum of consecutive primes in exactly 6 ways.

Original entry on oeis.org

34421, 130638, 229841, 235493, 271919, 295504, 345011, 347856, 358446, 358877, 414221, 429804, 434669, 480951, 488603, 532423, 532823, 543625, 561375, 621937, 626852, 655561, 687496, 703087, 734069, 746829, 810418, 824099, 888793

Offset: 1

Jud McCranie, May 30 2000

nonn

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054998, A054999, A055000.

A054845(a(n)) = 6. - Ray Chandler, Sep 20 2023

A307610 Number of partitions of prime(n) into consecutive primes.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 2, 3, 1, 1, 2, 1, 1, 3, 1, 2, 2, 2, 1, 3, 1, 1, 1, 5, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 3, 2, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 1, 1, 2

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

a(n) - 1 = number of partitions of prime(n) into two or more consecutive primes. - Ray Chandler, Sep 26 2023

			prime(13) = 41 = 2 + 3 + 5 + 7 + 11 + 13 = 11 + 13 + 17, so a(13) = 3.

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A000040, A034707, A054845, A056768, A070215, A084143.

a(n) = [x^prime(n)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^prime(k).

a(n) = A054845(A000040(n)).

A050940 Numbers that are not the sum (of a nonempty sequence) of consecutive primes.

Original entry on oeis.org

0, 1, 4, 6, 9, 14, 16, 20, 21, 22, 25, 27, 32, 33, 34, 35, 38, 40, 44, 45, 46, 50, 51, 54, 55, 57, 62, 63, 64, 65, 66, 69, 70, 74, 76, 80, 81, 82, 85, 86, 87, 91, 92, 93, 94, 96, 99, 104, 105, 106, 108, 110, 111, 114, 115, 116, 117, 118, 122, 123, 125

Offset: 1

N. J. A. Sloane, Jan 02 2000

nonn

Where is there a proof that this sequence is infinite? - Carlos Rivera, Apr 17 2002

Moser shows that the average order of A054845 is log(2), and hence this sequence is infinite with lower density at least 1 - log 2 = 0.306.... - Charles R Greathouse IV, Mar 21 2011

			The number 14 cannot be expressed as a sum of any consecutive subset of the following primes: {2, 3, 5, 7, 11, 13}.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001
Leo Moser, Notes on number theory. III. On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), pp. 159-161.

Complement of A034707.

10 N=1 20 N=N+1: if N=prmdiv(N) then goto 20 30 P=1 40 P=nxtprm(P):S=P:Q=P: if S>N\2 then print N;:goto 20 50 Q=nxtprm(Q):S=S+Q 60 if S=N then goto 20 70 if S>N then goto 40 80 goto 50

is(n)=if(isprime(n), return(0)); my(v,m=1,t); while(1, v=vector(m++); v[m\2]=precprime(n\m); for(i=m\2+1,m,v[i]=nextprime(v[i-1]+1)); forstep(i=m\2-1,1,-1,v[i]=precprime(v[i+1]-1)); if(v[1]==0, return(1)); t=vecsum(v); if (t==n, return(0)); if(t>n, while(t>n, t-=v[m]; v=concat(precprime(v[1]-1), v[1..m-1]); t+=v[1]), while(tCharles R Greathouse IV, May 05 2016

A054845(a(n)) = 0. - Ray Chandler, Sep 20 2023

cp's OEIS Frontend

A050936 Sum of two or more consecutive prime numbers.

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A054859 Smallest positive integer that can be expressed as the sum of consecutive primes in exactly n ways.

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A034707 Numbers that are sums (of a nonempty sequence) of consecutive primes.

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A054996 Integers that can be expressed as the sum of consecutive primes in exactly 1 way.

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A054998 Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

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A054999 Integers that can be expressed as the sum of consecutive primes in exactly 4 ways.

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A054997 Integers that can be expressed as the sum of consecutive primes in exactly 2 ways.

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A055001 Integers that can be expressed as the sum of consecutive primes in exactly 6 ways.

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