A307610 -id:A307610 - cp's OEIS frontend

A067377 Primes expressible as the sum of (at least two) consecutive primes in at least 1 way.

5, 17, 23, 31, 41, 53, 59, 67, 71, 83, 97, 101, 109, 127, 131, 139, 173, 181, 197, 199, 211, 223, 233, 251, 263, 269, 271, 281, 311, 331, 349, 353, 373, 379, 401, 421, 431, 439, 443, 449, 457, 463, 479, 487, 491, 499, 503, 523, 563, 587, 593, 607, 617, 631, 647, 659, 661, 677, 683, 691, 701, 719

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

			The prime 83, for example, is the sum of the five consecutive primes 11 + 13 + 17 + 19 + 23.
The prime 2011, for example, is the sum of the eleven consecutive primes 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211. - _Daniel Forgues_, Nov 03 2011

Hans Havermann, Table of n, a(n) for n = 1..34589
Patrick De Geest, WONplate 122
Hans Havermann, List of possible number of consecutive primes for n = 1..293768
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A050936, A067372-A067381, A307610.

Cf. A197227 (primes that are not the sum of consecutive primes).

p = {}; Do[a = Table[ Prime[i], {i, n, 150}]; l = Length[a]; k = 2; While[k < l + 1, b = Plus @@@ Partition[a, k]; k++; p = Append[ p, Select[ b, PrimeQ[ # ] &]]], {n, 1, 149}]; Take[ Union[ Flatten[p]], 70]
m=5!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[PrimeQ[p]&&p<=Prime[m]*3+8,AppendTo[lst,p]],{b,a+1,m+2,1}],{a,m}]; Union[lst] (* Vladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

Prime(n) such that A307610(n) > 1. - Ray Chandler, Sep 21 2023

Offset changed to 1 by Hans Havermann, Oct 07 2018

A067380 Primes expressible as the sum of (at least two) consecutive primes in at least 4 ways.

Original entry on oeis.org

311, 863, 14369, 14699, 15329, 19717, 29033, 34421, 36467, 37607, 40433, 42463, 48731, 49253, 49499, 55813, 67141, 70429, 76423, 78791, 85703, 90011, 94559, 97159, 98411, 109159, 110359, 110527, 125821, 130513, 134921, 141587, 147031, 147347, 155087, 155387

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

Note that the definition says "at least two", so a(n) = a(n) itself is not allowed as a possible sum (see Examples).

			311 is a term because 311 is prime and
  11+13+17+19+23+29+31+37+41+43+47 = 311,
  31+37+41+43+47+53+59 = 311,
  53+59+61+67+71 = 311,
  101+103+107 = 311.
1151 is not a term, since although 1151 is prime it only has three representations of the required form:
  101+97+89+83+79+73+71+67+61+59+53+47+43+41+37+31+29+23+19+17+13+11+7 = 1151,
  239+233+229+227+223 = 1151,
  389+383+379 = 1151.
Also, 16277 is not a term because although it has five representations as a sum of consecutive primes, it is not itself a prime. - _Sean A. Irvine_, Dec 25 2021

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
P. De Geest, WONplate 122
Sean A. Irvine, Java program (github)
C. Rivera, Puzzle 46

Cf. A050936, A067372-A067381, A307610.

M:=160000; P:=PrimesUpTo(M); S:=[0]; for p in P do t:=S[#S]+p; if #S ge 3 then if t-S[#S-2] gt M then break; end if; end if; S[#S+1]:=t;end for; c:=[0:j in [1..M]]; for C in [2..#S-1] do if IsEven(C) then L:=1; else L:=#S-C; end if; for j in [1..L] do s:=S[j+C]-S[j]; if s gt M then break; end if; if IsPrime(s) then c[s]+:=1; end if; end for; end for; [j:j in [1..M]|c[j] ge 4]; // Jon E. Schoenfield, Dec 25 2021

m=7!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[PrimeQ[p]&&pVladimir Joseph Stephan Orlovsky, Aug 15 2009 *)

Prime(n) such that A307610(n) > 4. - Ray Chandler, Sep 21 2023

The terms have been confirmed by Sean A. Irvine, Dec 24 2021. - N. J. A. Sloane, Dec 25 2021

A067378 Primes expressible as the sum of (at least two) consecutive primes in at least 2 ways.

Original entry on oeis.org

41, 83, 197, 199, 223, 251, 281, 311, 401, 439, 491, 593, 733, 857, 863, 883, 941, 983, 991, 1061, 1151, 1187, 1283, 1361, 1367, 1381, 1433, 1439, 1493, 1511, 1523, 1553, 1607, 1753, 1801, 1823, 1901, 1951, 2011, 2027, 2099, 2111, 2179, 2203, 2267, 2357, 2393, 2417, 2579, 2647, 2689, 2731

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

Jon E. Schoenfield, 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.

Cf. A050936, A067372-A067381, A307610.

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

e=2500; for(d=2,e, if(d%2==1,h=d/3,h=d/2); f=floor(2*d/(log(d)*3)); g=0; for(c=1,f,a=0; b=0; forprime(n=prime(c),h+50,a=a+n; b=b+1;if (a==d,g=g+1; if(g>=2&isprime(a),print1(a, ", ")),if(a>d,next(2)))))) /* The parameter g selects the number of ways wanted. - Robin Garcia, Jan 11 2011 */

Prime(n) such that A307610(n) > 2. - Ray Chandler, Sep 21 2023

A067379 Primes expressible as the sum of (at least two) consecutive primes in at least 3 ways.

Original entry on oeis.org

311, 863, 1151, 1367, 1951, 2393, 2647, 2689, 3389, 4957, 5059, 5153, 7451, 7901, 8819, 10499, 10859, 10949, 12329, 12641, 12713, 13127, 13297, 14369, 14699, 14759, 14951, 15091, 15329, 15527, 16223, 16249, 16829, 18089, 18311, 18401

Offset: 1

Patrick De Geest, Feb 04 2002

nonn

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
Patrick De Geest, WON plate 122
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A050936, A067372-A067381, A307610.

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

Prime(n) such that A307610(n) > 3. - Ray Chandler, Sep 21 2023

A197227 Primes that are not the sum of at least two consecutive primes.

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 29, 37, 43, 47, 61, 73, 79, 89, 103, 107, 113, 137, 149, 151, 157, 163, 167, 179, 191, 193, 227, 229, 239, 241, 257, 277, 283, 293, 307, 313, 317, 337, 347, 359, 367, 383, 389, 397, 409, 419, 433, 461, 467, 509, 521, 541, 547, 557, 569

Offset: 1

T. D. Noe, Nov 03 2011

nonn

Complement of A067377 in the primes. For the primes less than 10^6, these primes make up about 56%.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A050940, A067377, A307610.

P:= [seq(ithprime(i),i=1..10^3)]:
S:= ListTools:-PartialSums([0,op(P)]):
sort(convert(convert(P,set) minus {seq(seq(S[i]-S[j],j=1..i-2),i=1..10^3+1)},list)); # Robert Israel, May 09 2021

lim = 1000; pFound = {}; ps = Prime[Range[PrimePi[lim]]]; sm = ps; i = 0; While[i++; j = 1; While[sm[[j]] = sm[[j]] + ps[[i + j]]; sm[[j]] <= lim, If[PrimeQ[sm[[j]]], AppendTo[pFound, sm[[j]]]]; j++]; j > 1]; Complement[ps, pFound]

Prime(n) such that A307610(n) = 1. - Ray Chandler, Sep 21 2023

Definition clarified by Jonathan Sondow, May 18 2013

cp's OEIS Frontend

A067377 Primes expressible as the sum of (at least two) consecutive primes in at least 1 way.

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A067380 Primes expressible as the sum of (at least two) consecutive primes in at least 4 ways.

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A067378 Primes expressible as the sum of (at least two) consecutive primes in at least 2 ways.

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A067379 Primes expressible as the sum of (at least two) consecutive primes in at least 3 ways.

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A197227 Primes that are not the sum of at least two consecutive primes.

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