A050936 -id:A050936 - cp's OEIS frontend

A001043 Numbers that are the sum of 2 successive primes.

5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508, 520

Offset: 1

N. J. A. Sloane, R. K. Guy

Arithmetic derivative (see A003415) of prime(n)*prime(n+1). - Giorgio Balzarotti, May 26 2011
A008472(a(n)) = A191583(n). - Reinhard Zumkeller, Jun 28 2011
With the exception of the first term, all terms are even. a(n) is divisible by 4 if the difference between prime(n) and prime(n + 1) is not divisible by 4; e.g., prime(n) = 1 mod 4 and prime(n + 1) = 3 mod 4. In general, for a(n) to be divisible by some even number m > 2 requires that prime(n + 1) - prime(n) not be a multiple of m. - Alonso del Arte, Jan 30 2012

			2 + 3 = 5.
3 + 5 = 8.
5 + 7 = 12.
7 + 11 = 18.

Archimedeans Problems Drive, Eureka, 26 (1963), 12.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Albert Frank & Philippe Jacqueroux, International Contest, 2001. Item 22
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021)

Subsequence of A050936.

Cf. A000040 (primes), A031131 (first differences), A092163 (bisection), A100479 (bisection).

a001043 n = a001043_list !! (n-1)
a001043_list = zipWith (+) a000040_list $ tail a000040_list
-- Reinhard Zumkeller, Oct 19 2011

[(NthPrime(n+1) + NthPrime(n)): n in [1..100]]; // Vincenzo Librandi, Apr 02 2011

Primes:= select(isprime,[2,seq(2*i+1,i=1..1000)]):
n:= nops(Primes):
Primes[1..n-1] + Primes[2..n]; # Robert Israel, Aug 29 2014

Table[Prime[n] + Prime[n + 1], {n, 55}] (* Ray Chandler, Feb 12 2005 *)
Total/@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Aug 23 2011 *)
Abs[Differences[Table[(-1)^n Prime[n], {n, 60}]]] (* Alonso del Arte, Feb 03 2016 *)

p=2;forprime(q=3,1e3,print1(p+q", ");p=q) \\ Charles R Greathouse IV, Jun 10 2011

is(n)=precprime((n-1)/2)+nextprime(n/2)==n&&n>2 \\ Charles R Greathouse IV, Jun 21 2012

BB = primes_first_n(56)
L = []
for i in range(55): L.append(BB[1 + i] + BB[i])
L # Zerinvary Lajos, May 14 2007

a(n) = prime(n) + prime(n + 1) = A000040(n) + A000040(n+1).
a(n) = A116366(n, n - 1) for n > 1. - Reinhard Zumkeller, Feb 06 2006
a(n) = 2*A024675(n-1), n>1. - R. J. Mathar, Jan 12 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 17 2000

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

A067381 Smallest prime expressible as the sum of (at least two) consecutive primes in n ways.

Original entry on oeis.org

5, 41, 311, 311, 34421, 442019, 3634531, 48205429, 1798467197, 36154343837, 166400805323, 6123584726269

Offset: 1

Patrick De Geest, Feb 04 2002

a(9)-a(11) found by Wilfred Whiteside in 2007.

			5 can be represented (in just one way) as 2+3.

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-A067380, A054859.

a(12) from Giovanni Resta, May 07 2020

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

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

Original entry on oeis.org

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

A097889 Numbers that are products of (at least two) consecutive primes.

Original entry on oeis.org

6, 15, 30, 35, 77, 105, 143, 210, 221, 323, 385, 437, 667, 899, 1001, 1147, 1155, 1517, 1763, 2021, 2310, 2431, 2491, 3127, 3599, 4087, 4199, 4757, 5005, 5183, 5767, 6557, 7387, 7429, 8633, 9797, 10403, 11021, 11663, 12317, 12673, 14351, 15015, 16637, 17017

Offset: 1

Bart la Bastide (bart(AT)xs4all.nl), Sep 21 2004

Subsequence of A073485; A073490(a(n)) = 0. - Reinhard Zumkeller, Nov 20 2004
A proper subset of A073485. - Robert G. Wilson v, Jun 11 2010
A192280(a(n)) * (1 - A010051(a(n))) = 1. - Reinhard Zumkeller, Aug 26 2011 [corrected by Jason Yuen, Aug 29 2024]
The Heinz numbers of the partitions into at least 2 consecutive parts. The Heinz number of an integer partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Examples: (i) 105 (=3*5*7) is in the sequence because it is the Heinz number of the partition [2,3,4]; (ii) 108 (= 2*2*3*3*3) is not in the sequence because it is the Heinz number of the partition [1,1,2,2,2]. - Emeric Deutsch, Oct 02 2015

			1001 = 7 * 11 * 13.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Union of A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, etc.
Cf. A050936.
Intersection of A073485 and A002808.
Cf. A151800, A215366.

import Data.Set (singleton, deleteFindMin, insert)
a097889 n = a097889_list !! (n-1)
a097889_list = f $ singleton (6, 2, 3) where
   f s = y : f (insert (w, p, q') $ insert (w `div` p, a151800 p, q') s')
         where w = y * q'; q' = a151800 q
               ((y, p, q), s') = deleteFindMin s
-- Reinhard Zumkeller, May 12 2015, Aug 26 2011

isA097889 := proc(n)
    local plist,p,i ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    if nops(plist) < 2 then
        return false;
    end if;
    for i from 1 to nops(plist) do
        p := op(i,plist) ;
        if modp(n,p^2) = 0 then
            return false;
        end if;
        if i > 1 then
            if nextprime(op(i-1,plist)) <> p then
                return false;
            end if;
        end if;
    end do:
    true;
end proc:
for n from 1 to 1000 do
    if isA097889(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016

a = {}; Do[ AppendTo[a, Apply[ Times, (Prime /@ Partition[ Range[30], n, i]), 1]], {n, 2, 6}, {i, n - 1}]; Take[ Union[ Flatten[ a]], 45] (* Robert G. Wilson v, Sep 24 2004 *)

list(lim)=my(v=List(), p, t); for(e=2, log(lim+.5)\log(2), p=1; t=prod(i=1, e-1, prime(i)); forprime(q=prime(e), lim, t*=q/p; if(t>lim, next(2)); listput(v, t); p=nextprime(p+1))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Oct 24 2012

import heapq
from sympy import sieve
sieve.extend(10**6)
primes = list(sieve._list)
def prime(n): return primes[n-1]
def aupton(terms, verbose=False):
    p = prime(1)*prime(2); h = [(p, 1, 2)]; nextcount = 3; alst = []
    while len(alst) < terms:
        (v, s, l) = heapq.heappop(h)
        alst.append(v)
        if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} prime(i)]")
        if v >= p:
            p *= prime(nextcount)
            heapq.heappush(h, (p, 1, nextcount))
            nextcount += 1
        v //= prime(s); s += 1; l += 1; v *= prime(l)
        heapq.heappush(h, (v, s, l))
    return alst
print(aupton(45)) # Michael S. Branicky, Jun 15 2021

a(n) ~ n^2 log^2 n. - Charles R Greathouse IV, Oct 24 2012

More terms from Robert G. Wilson v, Sep 24 2004

Data corrected for n > 41 by Reinhard Zumkeller, Aug 26 2011

A055514 Composite numbers that are the sum of consecutive prime numbers and are divisible by the first and last of these primes.

Original entry on oeis.org

10, 39, 155, 371, 10225245560, 2935561623745, 454539357304421, 7228559051256366318, 1390718713078158117206

Offset: 1

Jud McCranie, Jul 03 2000

Composite n such that n = p_1 + p_2 + ... + p_k where the p_i are consecutive primes and n is divisible by p_1 and p_k.
Problem proposed by Carlos Rivera, who found the first 4 terms.
No more terms below 10^22. - Michael Beight, Jul 22 2012
In subsequence A055233 the first and last term of the sum must also be its smallest and largest prime factor. Therefore a(5) (cf. first EXAMPLE) is not in that sequence, since it has smaller factors 2^3*5. - M. F. Hasler, Nov 21 2021

			503 + 509 + 521 + ... + 508213 = 10225245560, which is divisible by 503 and 508213. - _Manuel Valdivia_, Nov 17 2011
From _Michael Beight_, Jul 22 2012: (Start)
a(8) = 7228559051256366318 = 73 + ... + 18281691653;
a(9) = 1390718713078158117206 = 370794889 + ... + 267902967061. (End)

C. Rivera, Puzzle

Subsequence of A050936.

Cf. A055233.

Module[{nn=200},Table[Total/@Select[Partition[Prime[Range[10000]],n,1],scpQ],{n,2,nn}]]//Flatten (* The program generates the first four terms of the sequence. *)
(* Harvey P. Dale, Oct 22 2022 *)

S=vector(N=50000); s=0; i=1; forprime(p=2,oo, S[i++]=s+=p; for(j=1,i-2, (s-S[j])%p || (s-S[j])%prime(j)|| print1(s-S[j]",")|| break)) \\ gives a(1..5), but too slow to go beyond. - M. F. Hasler, Nov 21 2021

a(7) from Donovan Johnson, Jun 19 2008

a(8) and a(9) from Michael Beight, Jul 22 2012

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

A342443 a(n) is the largest prime < 10^n that is the sum of at least two consecutive primes.

Original entry on oeis.org

5, 97, 991, 9949, 99971, 999983, 9999991, 99999989, 999999937, 9999999943, 99999999977, 999999999989, 9999999999763, 99999999999959, 999999999999989

Offset: 1

Bernard Schott, Mar 12 2021

The minimum corresponding number of consecutive primes to get this largest prime a(n) is A342444(n) and the first prime of these A342444(n) consecutive primes is A342454(n).
Differs from A342439 where the corresponding primes result of the longest sum < 10^n of consecutive primes.
a(n) is the largest n-digit prime A003618(n) for n = 2, 6, 7, 8, 9, 11, 12, ...
a(13) >= k = 10^13 - 237. If a(13) > k then it is the sum of at least 30000 primes. k can be written as the sum of 6449 consecutive primes. - David A. Corneth, Mar 13 2021
No sum of 30000 or more consecutive primes is in the interval [10^13 - 237, 10^13 - 1], so a(13) = 10^13 - 237. - Jon E. Schoenfield, Mar 14 2021

			a(1) = 5 = 2 + 3, since it is not possible to obtain the greatest 1-digit prime 7 when adding consecutive primes.
a(2) = 29 + 31 + 37 = 97, since (29, 31, 37) are consecutive primes and 97 is the largest 2-digit prime.

Cf. A003618, A050936, A067377, A342439, A342440, A342444, A342453, A342454.

A342439(n) <= a(n) <= A003618(n).

a(9) from Jinyuan Wang, Mar 13 2021
a(10) from David A. Corneth, Mar 13 2021
a(11)-a(12) from Jinyuan Wang, Mar 13 2021
a(13)-a(14) from Jon E. Schoenfield, Mar 13 2021
a(15) from Max Alekseyev, Dec 11 2024

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

cp's OEIS Frontend

A001043 Numbers that are the sum of 2 successive primes.

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

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A067381 Smallest prime expressible as the sum of (at least two) consecutive primes in n ways.

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

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A097889 Numbers that are products of (at least two) consecutive primes.

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