A034707 -id:A034707 - cp's OEIS frontend

A034961 Sums of three consecutive primes.

10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, 471, 487, 503, 519, 533, 551, 565, 581, 589, 607, 633, 661, 679, 689, 701, 713, 731, 749, 771

Offset: 1

Patrick De Geest, Oct 15 1998

For prime terms see A034962. - Zak Seidov, Feb 17 2011

			a(1) = 10 = 2 + 3 + 5.
a(42) = 565 = 181 + 191 + 193.

Zak Seidov, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 1021. p(k)+p(k+1)+1, The Prime Puzzles and Problems Connection.

Cf. A001043, A011974, A034707, A034962, A034963.

[&+[ NthPrime(n+k): k in [0..2] ]: n in [1..50] ]; // Vincenzo Librandi, Apr 03 2011

Plus @@@ Partition[ Prime[ Range[60]], 3, 1] (* Robert G. Wilson v, Feb 11 2005 *)
3 MovingAverage[Prime[Range[60]], {1, 1, 1}] (* Jean-François Alcover, Nov 12 2018 *)

a(n)=my(p=prime(n),q=nextprime(p+1)); p+q+nextprime(q+1) \\ Charles R Greathouse IV, Jul 01 2013

is(n)=my(p=precprime(n\3),q=nextprime(n\3+1),r=n-p-q); if(r>q, r==nextprime(q+2), r==precprime(p-1) && r) \\ Charles R Greathouse IV, Jul 05 2017

from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    p, q, r = 2, 3, 5
    while True:
        yield p + q + r
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 54))) # Michael S. Branicky, Dec 27 2022

BB = primes_first_n(57)
L = []
for i in range(55):
    L.append(BB[i]+BB[i+1]+BB[i+2])
L # Zerinvary Lajos, May 14 2007

a(n) = Sum_{k=0..2} A000040(n+k). - Omar E. Pol, Feb 28 2020

a(n) = A001043(n) + A000040(n+2). - R. J. Mathar, May 25 2020

A034963 Sums of four consecutive primes.

Original entry on oeis.org

17, 26, 36, 48, 60, 72, 88, 102, 120, 138, 152, 168, 184, 202, 220, 240, 258, 272, 290, 306, 324, 348, 370, 390, 408, 420, 432, 456, 480, 508, 534, 556, 576, 596, 620, 638, 660, 682, 700, 724, 744, 762, 780, 800, 830, 860, 890, 912, 928, 942, 964, 988, 1012

Offset: 1

Patrick De Geest, Oct 15 1998

			a(7) = 17 + 19 + 23 + 29 = 88.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A011974, A034707.

[&+[ NthPrime(n+k): k in [0..3] ]: n in [1..90] ]; /* Vincenzo Librandi, Apr 03 2011 */

A034963 := proc(n)
    add(ithprime(i), i=n..n+3) ;
end proc: # R. J. Mathar, Jun 06 2013

Plus@@@Partition[Prime[Range[6! ]],4,1] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)

a(n)=my(p=prime(n));p+sum(i=1,3,p=nextprime(p+1)) \\ Charles R Greathouse IV, Jul 01 2013

a(n) = Sum_{k=0..3} A000040(n+k). - Omar E. Pol, Mar 02 2020

A034962 Primes that are the sum of three consecutive primes.

Original entry on oeis.org

23, 31, 41, 59, 71, 83, 97, 109, 131, 173, 199, 211, 223, 251, 269, 311, 349, 439, 457, 487, 503, 607, 661, 701, 829, 857, 883, 911, 941, 1033, 1049, 1061, 1151, 1187, 1229, 1249, 1303, 1367, 1381, 1409, 1433, 1493, 1511, 1553, 1667, 1867, 1931, 1973, 1993

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

Or, primes in A034961 (Sums of three consecutive primes). - Zak Seidov, Feb 16 2011

			E.g., 131 = 41 + 43 + 47.
A034962(n) = p+q+r, where p = A073681(n), and p<q<r are three consecutive primes. - _Zak Seidov_, Mar 09 2009

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Table of n, primepi(A073681(n)), primepi(A034962(n)), A073681(n), A152469(n), A152470(n), A034962(n) for n = 1..10000
Zak Seidov, Table of a(1000*k) for k=1..1346.

Cf. A001043, A011974, A034707, A034961. Different from A050207.

Cf. A073681 (smallest of three consecutive primes whose sum is a prime).

[a: n in [1..150] | IsPrime(a) where a is NthPrime(n)+NthPrime(n+1)+NthPrime(n+2)]; // Vincenzo Librandi, Jun 23 2016

a:=proc(n) if isprime(ithprime(n)+ithprime(n+1)+ithprime(n+2))=true then ithprime(n)+ithprime(n+1)+ithprime(n+2) else fi end: seq(a(n), n=1..120); # Emeric Deutsch, Apr 24 2006

a = {}; Do[k = Prime[x] + Prime[x + 1] + Prime[x + 2]; If[PrimeQ[k], AppendTo[a, k]], {x, 1, 350}]; a (* Artur Jasinski, Jan 27 2007 *)
Select[(Plus@@@Partition[Prime[Range[200]],3,1]),PrimeQ] (* Zak Seidov, Feb 07 2012 *)
Select[ListConvolve[{1,1,1},Prime[Range[200]]],PrimeQ] (* Harvey P. Dale, Jul 12 2013 *)

forprime(p=2,1000, p2=nextprime(p+1); p3=nextprime(p2+1); q=p+p2+p3; if(isprime(q),print1(q",")) ) \\ Max Alekseyev, Jan 26 2007

{p=2;q=3;for(n=1,100,r=nextprime(q+1); if(isprime(t=p+q+r),print1(t","));p=q;q=r;)} \\ Zak Seidov, Mar 09 2009

from itertools import count, islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    p, q, r = 2, 3, 5
    while True:
        if isprime(p+q+r): yield p+q+r
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 50))) # Michael S. Branicky, Dec 27 2022

A050936 Sum of two or more consecutive prime numbers.

Original entry on oeis.org

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

A034964 Sums of five consecutive primes.

Original entry on oeis.org

28, 39, 53, 67, 83, 101, 119, 139, 161, 181, 199, 221, 243, 263, 287, 311, 331, 351, 373, 395, 421, 449, 473, 497, 517, 533, 559, 587, 617, 647, 683, 707, 733, 759, 787, 811, 839, 863, 891, 917, 941, 961, 991, 1023, 1057, 1089, 1123, 1151, 1169, 1193

Offset: 1

Patrick De Geest, Oct 15 1998

Except for the first term, all terms are odd. - Alonso del Arte, Dec 30 2011

			a(1) = prime(1+0) + prime(1+1) + prime(1+2) + prime(1+3) + prime(1+4) = 2 + 3 + 5 + 7 + 11 = 28.
a(2) = prime(2+0) + prime(2+1) + prime(2+2) + prime(2+3) + prime(2+4) = 3 + 5 + 7 + 11 + 13 = 39.

Owen O'Shea and Underwood Dudley, The Magic Numbers of the Professor, Mathematical Association of America (2007), p. 62

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A011974, A034707.

Cf. A131686 (sums of five consecutive squares of primes).

[&+[ NthPrime(n+k): k in [0..4] ]: n in [1..100] ]; // Vincenzo Librandi, Apr 03 2011

A034964:=n->add(ithprime(i), i=n..n+4): seq(A034964(n), n=1..50); # Wesley Ivan Hurt, Sep 14 2014

Plus@@@Partition[Prime[Range[100]],5,1] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)

a(n) = sum(k=n, n+4, prime(k)); \\ Michel Marcus, Sep 03 2016

first(n) = {my(psum = 28, pr = List([2,3,5,7,11]), res = List([28])); for(i=2,n, psum -= pr[1]; listpop(pr, 1); listput(pr, nextprime(pr[4] + 1)); psum += pr[5]; listput(res, psum)); res} \\ David A. Corneth, Oct 14 2017

BB = primes_first_n(60)
L = []
for i in range(55):
    L.append(BB[i]+BB[i+1]+BB[i+2]+BB[i+3]+BB[i+4])
L # Zerinvary Lajos, May 14 2007

a(n) = Sum_{i=n..n+4} prime(i). - Wesley Ivan Hurt, Sep 14 2014

Offset changed to 1 by Joerg Arndt, Sep 04 2016

A054643 Primes prime(n) such that prime(n) + prime(n+1) + prime(n+2) == 0 (mod 3).

Original entry on oeis.org

3, 47, 151, 167, 199, 251, 257, 367, 503, 523, 557, 587, 601, 647, 727, 941, 971, 991, 1063, 1097, 1117, 1181, 1217, 1231, 1361, 1453, 1493, 1499, 1531, 1741, 1747, 1753, 1759, 1889, 1901, 1907, 2063, 2161, 2281, 2393, 2399, 2411, 2441, 2671, 2897, 2957

Offset: 1

Labos Elemer, May 15 2000

nonn

The 2 differences of these 3 primes should be congruent of 6, except the first prime 3, for which 3 + 5 + 7 = 15 holds. Sequences like A047948, A052198 etc. are subsequences here.

			For prime(242) = 1531, the sum is 4623, the mean is 1541 and the successive differences are 6a=12 or 6b=6 resp.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A034961, A034707, A024675, A052288, A047948, A052198.
A122535 is a subsequence.
Cf. A075541 (for their indices).

Select[Partition[Prime@ Range@ 430, 3, 1], Divisible[Total@ #, 3] &][[All, 1]] (* Michael De Vlieger, Jun 29 2017 *)

A034965 Primes that are sum of five consecutive primes.

Original entry on oeis.org

53, 67, 83, 101, 139, 181, 199, 263, 311, 331, 373, 421, 449, 587, 617, 647, 683, 733, 787, 811, 839, 863, 941, 991, 1123, 1151, 1193, 1361, 1381, 1579, 1609, 1801, 1831, 1861, 1949, 1979, 2081, 2113, 2143, 2221, 2273, 2297, 2357, 2423, 2459, 2689, 2731

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			53 = 5 + 7 + 11 + 13 + 17.
373 = 67 + 71 + 73 + 79 + 83.

Harvey P. Dale and Syed Iddi Hasan, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A001043, A011974, A034707, A152468. Also Cf. A034964, of which this sequence is a subset.

ts_prod_n:=proc(n) local i,ans; ans:=[ ]: for i from 1 to n do if isprime(ithprime(i)+ithprime(i+1)+ithprime(i+2)+ithprime(i+3)+ithprime(i+4))= 'true' then ans:=[op(ans), ithprime(i)+ithprime(i+1)+ithprime(i+2)+ithprime(i+3)+ithprime(i+4) ]: fi od: end: ts_prod_n(701); # Jani Melik, May 05 2006

Select[Table[Plus@@Prime[Range[n, n + 4]], {n, 200}], PrimeQ] (* Alonso del Arte, Dec 30 2011 *)
Select[Total/@Partition[Prime[Range[200]],5,1],PrimeQ] (* Harvey P. Dale, May 24 2012 *)

Corrected example by Paul S. Coombes, Dec 29 2011

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

A340771 Numbers which are the sum of some number of consecutive prime squares.

Original entry on oeis.org

4, 9, 13, 25, 34, 38, 49, 74, 83, 87, 121, 169, 170, 195, 204, 208, 289, 290, 339, 361, 364, 373, 377, 458, 529, 579, 628, 650, 653, 662, 666, 819, 841, 890, 940, 961, 989, 1014, 1023, 1027, 1179, 1348, 1369, 1370, 1469, 1518, 1543, 1552, 1556, 1681, 1731, 1802, 1849

Offset: 1

Michel Marcus, Jan 20 2021

nonn

			The initial terms are 2^2, 3^2, 2^2+3^2, 5^2, 3^2+5^2, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Cathal O'Sullivan, Jonathan P. Sorenson, and Aryn Stahl, An Algorithm to Find Sums of Consecutive Powers of Primes, arXiv:2204.10930 [math.NT], 2022-2023. See S2 p. 10.
Janyarak Tongsomporn, Saeree Wananiyakul, and Jörn Steuding, Sums of Consecutive Prime Squares, arXiv:2101.07558 [math.NT], 2021.
Janyarak Tongsomporn, Saeree Wananiyakul, and Jörn Steuding, Sums of Consecutive Prime Squares, #A9 INTEGERS 22 (2022).

Cf. A001248, A034707.

N:= 10000: # for terms <= N
PS:= [0,seq(ithprime(i)^2,i=1..numtheory:-pi(floor(sqrt(N))))]:
SPS:= ListTools:-PartialSums(PS):
sort(convert(select(`<=`,{seq(seq(SPS[t]-SPS[s],s=1..t-1),t=2..nops(SPS))},N),list)); # Robert Israel, Jan 20 2021

lista(nn) = {my(list = List(), ip = primepi(nn), vp = primes(ip)); for(i=1, ip, my(s=vp[i]^2); listput(list, s); for (j=i+1, ip, s += vp[j]^2; if (s >vp[ip]^2, break); listput(list, s););); Vec(vecsort(list,,8));} \\ Michel Marcus, Jan 20 2021

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A034961 Sums of three consecutive primes.

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A034963 Sums of four consecutive primes.

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A034962 Primes that are the sum of three consecutive primes.

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A050936 Sum of two or more consecutive prime numbers.

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A034964 Sums of five consecutive primes.

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A054643 Primes prime(n) such that prime(n) + prime(n+1) + prime(n+2) == 0 (mod 3).

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