A084143 -id:A084143 - 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

A067372 Integers expressible as the sum of (at least two) consecutive primes in at least 2 ways.

Original entry on oeis.org

36, 41, 60, 72, 83, 90, 100, 112, 119, 120, 138, 143, 152, 180, 187, 197, 199, 204, 210, 221, 223, 228, 240, 251, 258, 276, 281, 287, 300, 304, 311, 323, 330, 340, 371, 372, 384, 390, 395, 401, 408, 410, 434, 439, 456, 462, 473, 480, 491, 492, 508, 510, 533

Offset: 1

Patrick De Geest, Feb 04 2002

			36 = (17 + 19) = (5 + 7 + 11 + 13) or (#2,17) (#4,5).

David A. Corneth, Table of n, a(n) for n = 1..10841 (terms <= 10^5, first 1000 terms from Donovan Johnson)
P. De Geest, WONplate 122
C. Rivera, Puzzle 46
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A050936, A067372-A067381, A054997.

m=5!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

upto(n) = {my(s = 0, pr = List([0]), l = List(), res = List()); forprime(p = 2, n + 100, s+=p; listput(pr, s) ); for(i = 3, #pr, for(j = 2, i-1, if(pr[i] - pr[i-j] <= n, listput(l, pr[i] - pr[i-j]) , next(2) ) ) ); listsort(l); for(i = 2, #l, if(l[i-1] == l[i], listput(res, l[i]) ) ); Set(res); } \\ David A. Corneth, Aug 22 2019

A084143(a(n)) > 1. - Ray Chandler, Sep 20 2023

Offset corrected by Donovan Johnson, Nov 14 2013

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)).

A084146 Integers that have exactly one representation as a sum of two or more consecutive primes.

Original entry on oeis.org

5, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 39, 42, 48, 49, 52, 53, 56, 58, 59, 67, 68, 71, 75, 77, 78, 84, 88, 95, 97, 98, 101, 102, 109, 121, 124, 127, 128, 129, 131, 132, 139, 144, 150, 155, 156, 158, 159, 160, 161, 162, 168, 169, 172, 173, 181, 184, 186, 192

Offset: 1

Eric W. Weisstein, May 15 2003

nonn

More fundamental than A050936, which gives integers having 1 *or more* such representations

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A050936, A084143, A337095 (subset of primes).

# uses code of A084143
isA084146 := proc(n::integer)
    if A084143(n) = 1 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 300 do
    if isA084146(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Aug 19 2020

A084143(a(n)) = 1. - Ray Chandler, Sep 20 2023

More terms from Matthew Conroy, May 25 2003

A067373 Integers expressible as the sum of (at least two) consecutive primes in at least 3 ways.

Original entry on oeis.org

240, 287, 311, 340, 371, 510, 660, 803, 863, 864, 931, 961, 990, 1012, 1060, 1099, 1104, 1151, 1164, 1236, 1313, 1320, 1367, 1392, 1524, 1643, 1650, 1710, 1788, 1793, 1854, 1951, 1956, 2040, 2303, 2304, 2387, 2393, 2436, 2507, 2556, 2586, 2647, 2670

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			240 = (113 + 127) = (53 + 59 + 61 + 67) = (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43) or (#2,113) (#4,53) (#8,17).

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. De Geest, WONplate 122
C. Rivera, Puzzle 46
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A050936, A067372-A067381, A054998.

m=3*5!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

A084143(a(n)) > 2. - Ray Chandler, Sep 20 2023

Offset corrected by Donovan Johnson, Nov 14 2013

A067374 Integers expressible as the sum of (at least two) consecutive primes in at least 4 ways.

Original entry on oeis.org

311, 863, 1164, 1320, 1650, 1854, 2856, 2867, 3198, 3264, 3754, 4200, 4920, 5100, 5770, 5999, 6504, 8152, 10134, 10320, 10536, 10649, 11058, 12294, 12438, 12762, 12820, 12954, 12990, 14369, 14699, 14826, 15329, 15610, 15762, 16199, 16277

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			E.g., 311 = 101 + 103 + 107 = 53 + 59 + 61 + 67 + 71 = 31 + 37 + 41 + 43 + 47 + 53 + 59 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47.

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. De Geest, WONplate 122
C. Rivera, Puzzle 46

Cf. A050936, A067372-A067381, A054999.

Clear[lst,lst1,m,n,p,a,b] m=2*6!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

A084143(a(n)) > 3. - Ray Chandler, Sep 20 2023

Offset corrected by Donovan Johnson, Nov 14 2013

A067375 Integers expressible as the sum of (at least two) consecutive primes in at least 5 ways.

Original entry on oeis.org

16277, 20272, 25416, 28500, 34421, 41074, 45101, 46660, 50560, 53424, 59068, 68787, 70104, 70692, 71548, 78756, 85433, 85481, 88453, 94350, 98881, 105827, 117907, 120151, 121847, 125952, 130638, 130789, 131420, 132539, 133367, 134376, 135918, 139853, 158810

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			E.g. 16277 = (#7,2297) (#11,1451) (#13,1213) (#35,359) (#37,331).

Donovan Johnson, Table of n, a(n) for n = 1..1000
P. De Geest, WONplate 122
C. Rivera, Puzzle 46

Cf. A050936, A067372-A067381, A055000.

t={};Do[p=Prime[m];Do[p=p+Prime[n];If[p<200000,AppendTo[t,p]],{n,m+1,7001}],{m,1,7000}];t=Sort@t;f5[l_]:=Module[{t={}},Do[If[l[[n]]==l[[n+4]],AppendTo[t,l[[n]]]],{n,Length[l]-4}];t];Union@f5[t] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

A084143(a(n)) > 4. - Ray Chandler, Sep 20 2023

Offset and a(35) corrected by Donovan Johnson, Nov 14 2013

A084147 Integers that have exactly 2 representations as sums of consecutive primes.

Original entry on oeis.org

36, 41, 60, 72, 83, 90, 100, 112, 119, 120, 138, 143, 152, 180, 187, 197, 199, 204, 210, 221, 223, 228, 251, 258, 276, 281, 300, 304, 323, 330, 372, 384, 390, 395, 401, 408, 410, 434, 439, 456, 462, 473, 480, 491, 492, 508, 533, 540, 551, 552, 558, 559, 576

Offset: 1

Eric W. Weisstein, May 15 2003

nonn

More fundamental than A067372, which gives integers having 2 *or more* such representations

			36 is in the sequence because it can be written in exactly two ways as sum of consecutive primes: 17+19 and 5+7+11+13.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A067372, A084143.

g:=sum(sum(product(x^ithprime(k),k=i..j),j=i+1..150),i=1..150): gser:=series(g,x=0,605): a:=proc(n) if coeff(gser,x^n)=2 then op(2,x^n) else fi end: seq(a(n),n=1..600); # Emeric Deutsch, Mar 30 2006
# Alternative
N:= 70: # for terms up to prime(N-1)+prime(N)
P:= [seq(ithprime(i),i=1..N)]: m:= P[N-1]+P[N]:
S:= ListTools:-PartialSums(P):
V:= Vector(m):
for i from 2 while S[i] <= m do V[S[i]]:= 1 od:
for i from 1 to N-2 do
  for j from i+2 to N while S[j]-S[i] <= m do V[S[j]-S[i]]:= V[S[j]-S[i]] + 1
od od:
select(t -> V[t] = 2, [$1..m]); # Robert Israel, Feb 14 2021

With[{nn=100},Take[Sort[Select[Tally[Flatten[Table[Total/@Partition[Prime[Range[nn]],n,1],{n,2,nn}]]],#[[2]]==2&]][[All,1]],nn]] (* Harvey P. Dale, Mar 06 2020 *)

More terms from John W. Layman, May 21 2003

A087072 Numbers having no partitions into a sum of two or more consecutive primes.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 13, 14, 16, 19, 20, 21, 22, 25, 27, 29, 32, 33, 34, 35, 37, 38, 40, 43, 44, 45, 46, 47, 50, 51, 54, 55, 57, 61, 62, 63, 64, 65, 66, 69, 70, 73, 74, 76, 79, 80, 81, 82, 85, 86, 87, 89, 91, 92, 93, 94, 96, 99, 103, 104, 105, 106, 107, 108, 110, 111

Offset: 1

Reinhard Zumkeller, Aug 08 2003

nonn

A084143(a(n)) = 0; complement of A050936.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A050936, A084143.

A163246 Squares which can be represented as the sum of consecutive primes in more than one way.

Original entry on oeis.org

36, 100, 576, 841, 961, 1764, 1849, 2209, 2304, 7056, 22801, 24649, 25600, 30276, 31684, 32400, 36481, 39601, 40000, 47524, 48400, 48841, 57600, 58081, 66564, 69169, 69696, 76729, 77284, 80089, 85849, 93636, 94864, 96721, 112896, 119716, 128164, 134689, 138384, 140625, 142884, 147456

Offset: 1

Gaurav Kumar, Jul 23 2009

nonn

			6^2 = 36 = 5 + 7 + 11 + 13 = 17 + 19.

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A000290, A034707, A050936, A067372, A084143, A163244.

{ A000290 } intersect { A034707 }.
{ A000290 } intersect { A050936 }.
k^2 such that A084143(k^2) > 1. - Georg Fischer, Jul 08 2022

Offset corrected by Arkadiusz Wesolowski, Mar 28 2012

Duplicate 2304 removed and some missing terms inserted by Georg Fischer, Jul 08 2022

cp's OEIS Frontend

A050936 Sum of two or more consecutive prime numbers.

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A067372 Integers expressible as the sum of (at least two) consecutive primes in at least 2 ways.

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

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A084146 Integers that have exactly one representation as a sum of two or more consecutive primes.

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A067373 Integers expressible as the sum of (at least two) consecutive primes in at least 3 ways.

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A067374 Integers expressible as the sum of (at least two) consecutive primes in at least 4 ways.

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A067375 Integers expressible as the sum of (at least two) consecutive primes in at least 5 ways.

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