A127341 -id:A127341 - cp's OEIS frontend

A215991 Primes that are the sum of 25 consecutive primes.

1259, 1361, 2027, 2267, 2633, 3137, 3389, 4057, 5153, 6257, 6553, 7013, 7451, 7901, 9907, 10499, 10799, 10949, 11579, 12401, 14369, 15013, 15329, 17377, 17903, 18251, 18427, 19309, 22441, 24023, 25057, 25229, 26041, 26699, 28111, 29017, 29207, 30707, 32939, 35051, 36583

Offset: 1

Syed Iddi Hasan, Aug 30 2012

nonn

Such sequences already existed for all odd numbers <= 15. I chose the particular points (in A215991-A216020) so that by referring to a particular n-th term of one of these sequences, the expected range of the n-th term of an x-prime sum can be calculated for any odd x<100000.

Syed Iddi Hasan, Table of n, a(n) for n = 1..10000

Cf. A034962, A034965, A082246, A082251, A127340, A127341, A161612, A215992, A215993, A215994, A215995, A215996, A215997, A215998, A215999, A216000, A216001, A216002, A216003, A216004, A216005, A216006, A216007, A216008, A216009, A216010, A216011, A216012, A216013, A216014, A216015, A216016, A216017, A216018, A216019, A216020.

P:=Filtered([1..10^4],IsPrime);;
Filtered(List([0..250],k->Sum([1..25],i->P[i+k])),IsPrime); # Muniru A Asiru, Feb 11 2018

select(isprime, [seq(add(ithprime(i+k), i=1..25), k=0..250)]); # Muniru A Asiru, Feb 11 2018

Select[ListConvolve[Table[1, 25], Prime[Range[500]]], PrimeQ] (* Jean-François Alcover, Jul 01 2018, after Harvey P. Dale *)
Select[Total/@Partition[Prime[Range[300]],25,1],PrimeQ] (* Harvey P. Dale, Mar 04 2023 *)

psumprm(m, n)={my(list=List(), s=sum(j=1,m,prime(j)), i=1); while(#listAndrew Howroyd, Feb 11 2018

A070934 Smallest prime equal to the sum of 2n+1 consecutive primes.

Original entry on oeis.org

2, 23, 53, 197, 127, 233, 691, 379, 499, 857, 953, 1151, 1259, 1583, 2099, 2399, 2417, 2579, 2909, 3803, 3821, 4217, 4651, 5107, 5813, 6829, 6079, 6599, 14153, 10091, 8273, 10163, 9521, 12281, 13043, 11597, 12713, 13099, 16763, 15527, 16823, 22741

Offset: 0

Lekraj Beedassy, May 21 2002

nonn

			Every term of the increasing sequence of primes 127, 401, 439, 479, 593,... is splittable into a sum of 9 consecutive odd primes and 127 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 is the least one corresponding to n = 4.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. Bisection of A070281.
See A082244 for another version.
Cf. A089793, A215235.
Cf. A034962, A082246, A082251, A127340, A127341, A161612, A215991-A216020.

f[n_] := Block[{k = 1, s},While[s = Sum[Prime[i], {i, k, k + 2n}]; !PrimeQ[s], k++ ]; s]; Table[f[n], {n, 0, 41}] (* Ray Chandler, Sep 27 2006 *)

Corrected and extended by Ray G. Opao, Aug 26 2004

Entry revised by Ray Chandler, Sep 27 2006

A127340 Primes that are the sum of 11 consecutive primes.

Original entry on oeis.org

233, 271, 311, 353, 443, 491, 631, 677, 883, 1367, 1423, 1483, 1543, 1607, 1787, 1901, 1951, 2011, 2141, 2203, 2383, 3253, 3469, 3541, 3617, 3691, 3967, 4159, 4229, 4297, 4943, 5009, 5483, 5657, 5741, 5903, 5981, 6553, 6871, 6991, 7057, 7121, 7187, 7873

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

Primes in A127338.

A prime number n is in the sequence if for some k it is the absolute value of coefficient of x^10 of the polynomial Prod_{j=0,10}(x-prime(k+j)); the roots of this polynomial are prime(k), ..., prime(k+10).

Syed Iddi Hasan, Table of n, a(n) for n = 1..10000

Cf. A127338, A034962, A034965, A082246, A082251, A127341.

a = {}; Do[If[PrimeQ[Sum[Prime[x + n], {n, 0, 10}]], AppendTo[a, Sum[Prime[x + n], {n, 0, 10}]]], {x, 1, 500}]; a
Select[Total/@Partition[Prime[Range[200]],11,1],PrimeQ] (* Harvey P. Dale, Jul 16 2012 *)

{m=125;k=11;for(n=0,m-1,a=sum(j=1,k,prime(n+j));if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 13 2007

{m=126;k=11;for(n=1,m,a=abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1));if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 13 2007

Edited by Klaus Brockhaus, Jan 13 2007

A127346 Primes in A127345.

Original entry on oeis.org

31, 71, 167, 311, 1151, 3119, 4871, 5711, 6791, 14831, 24071, 33911, 60167, 79031, 101159, 106367, 115631, 158231, 235751, 259751, 366791, 402551, 455471, 565919, 635711, 644951, 1124831, 1347971, 1510799, 1547927, 1743419, 1851671, 2048471

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

Primes of the form prime(k)*prime(k+1) + prime(k)*prime(k+2) + prime(k+1)*prime(k+2).

A prime number n is in the sequence if for some k it is the coefficient of x^1 of the polynomial Product_{j=0..2} (x-prime(k+j)); the roots of this polynomial are prime(k), ..., prime(k+2).

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

Cf. A127345, A127347, A127351, A006094, A002110, A034962, A034965, A082246, A082251, A127340, A127341, A070934, A046301, A046302, A046303, A046324, A046325, A046326, A046327.

b = {}; a = {}; Do[If[PrimeQ[Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[a, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]], AppendTo[b, Prime[x] Prime[x + 1] + Prime[x] Prime[x + 2] + Prime[x + 1] Prime[x + 2]]], {x, 1, 100}]; Print[a] (* Artur Jasinski, Jan 11 2007 *)
s[li_] := li[[1]]*(li[[2]]+li[[3]])+li[[2]]*li[[3]]; Select[(s[#]&/@Partition[Prime[Range[100]], 3, 1]), PrimeQ] (* Zak Seidov, Jan 13 2012 *)

{m=143;k=2;for(n=1,m,a=sum(i=n,n+k-1,sum(j=i+1,n+k,prime(i)*prime(j)));if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

{m=143;k=2;for(n=1,m,a=polcoeff(prod(j=0,k,(x-prime(n+j))),1);if(isprime(a),print1(a,",")))} \\ Klaus Brockhaus, Jan 21 2007

p=2; q=3; forprime(r=5, 1e3, if(isprime(t=p*q+p*r+q*r), print1(t", ")); p=q; q=r) \\ Charles R Greathouse IV, Jan 13 2012

a(n) = A127345(A204231(n)). - Zak Seidov, Jan 13 2012

Edited and extended by Klaus Brockhaus, Jan 21 2007

A082244 Smallest odd prime that is the sum of 2n+1 consecutive primes.

Original entry on oeis.org

3, 23, 53, 197, 127, 233, 691, 379, 499, 857, 953, 1151, 1259, 1583, 2099, 2399, 2417, 2579, 2909, 3803, 3821, 4217, 4651, 5107, 5813, 6829, 6079, 6599, 14153, 10091, 8273, 10163, 9521, 12281, 13043, 11597, 12713, 13099, 16763, 15527, 16823, 22741

Offset: 0

Cino Hilliard, May 09 2003

			For n = 2,
2+3+5+7+11=28
3+5+7+11+13=39
5+7+11+13+17=53
so 53 is the first prime that is the sum of 5 consecutive primes

Robert Israel, Table of n, a(n) for n = 0..10000

See A070934 for another version.

Cf. A034962, A082246, A082251, A127340, A127341, A161612, A215991-A216020.

P:= select(isprime, [seq(i,i=3..3000,2)]):
S:= [0,op(ListTools:-PartialSums(P))]: nS:= nops(S):
R:= NULL:
for n from 1 do
  found:= false;
  for j from 1 to nS - 2*n + 1 while not found do
    v:= S[j+2*n-1]-S[j];
    if isprime(v) then R:= R,v; found:= true fi
  od;
  if not found then break fi;
od:
R; # Robert Israel, Jan 09 2025

Join[{3},Table[SelectFirst[Total/@Partition[Prime[Range[1000]],2n+1,1],PrimeQ],{n,50}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 15 2016 *)

\\ First prime that the sum of an odd number of consecutive primes
psumprm(n) = { sr=0; forstep(i=1,n,2, s=0; for(j=1,i, s+=prime(j); ); for(x=1,n, s = s - prime(x)+ prime(x+i); if(isprime(s),sr+=1.0/s; print1(s" "); break); ); ); print(); print(sr) }

The sum of the reciprocals = 0.4304...

A342439 Let S(n,k) denote the set of primes < 10^n which are the sum of k consecutive primes, and let K = maximum k >= 2 such that S(n,k) is nonempty; then a(n) = max S(n,K).

Original entry on oeis.org

5, 41, 953, 9521, 92951, 997651, 9964597, 99819619, 999715711, 9999419621, 99987684473, 999973156643, 9999946325147, 99999863884699, 999999149973119, 9999994503821977, 99999999469565483, 999999988375776737, 9999999776402081701

Offset: 1

Bernard Schott, Mar 12 2021

Inspired by the 50th problem of Project Euler (see link).
There must be at least two consecutive primes in the sum.
The corresponding number K of consecutive primes to get this largest prime is A342440(n) and the first prime of these A342440(n) consecutive primes is A342453(n).
It can happen that the sums of K = A342440(n) consecutive primes give two (or more) distinct n-digit primes. In that case, a(n) is the greatest of these primes. Martin Ehrenstein proved that there are only two such cases when 1 <= n <= 19, for n = 7 and n = 15 (see corresponding examples).
Solutions and Python program are proposed in Dreamshire and Archive.today links. - Daniel Suteu, Mar 12 2021

			a(1) = 5 = 2+3.
a(2) = 41 = 2 + 3 + 5 + 7 + 11 + 13; note that 97 = 29 + 31 + 37 is prime, sum of 3 consecutive primes, but 41 is obtained by adding 6 consecutive primes, so, 97 is not a term.
A342440(7) = 1587, and there exist two 7-digit primes that are sum of 1587 consecutive primes; as 9951191 = 5+...+13399 < 9964597 = 7+...+13411 hence a(7) = 9964597.
A342440(15) = 10695879 , and there exist two 15-digit primes that are sum of 10695879 consecutive primes; as 999998764608469 = 7+...+192682309 < 999999149973119 = 13+...+192682337, hence a(15) = 999999149973119.

Dreamshire, Project Euler 50 Solution.
Archive.today, trizen / experimental-projects.
Project Euler, Problem 50: Consecutive prime sum.

Cf. A034962, A034965, A082246, A082251, A127340, A127341, A215991, A067377.

Cf. A342440, A342443, A342444, A342453, A342454.

Name improved by N. J. A. Sloane, Mar 12 2021
a(4)-a(17) from Daniel Suteu, Mar 12 2021
a(18)-a(19) from Martin Ehrenstein, Mar 13 2021
a(7) and a(15) corrected by Martin Ehrenstein, Mar 15 2021

A127492 Indices m of primes such that Sum_{k=0..2, k

Original entry on oeis.org

2, 10, 17, 49, 71, 72, 75, 145, 161, 167, 170, 184, 244, 250, 257, 266, 267, 282, 286, 301, 307, 325, 343, 391, 405, 429, 450, 537, 556, 561, 584, 685, 710, 743, 790, 835, 861, 904, 928, 953

Offset: 1

Artur Jasinski, Jan 16 2007

Let p_0 .. p_4 be five consecutive primes, starting with the m-th prime. The index m is in the sequence if the absolute value [x^0] of the polynomial (x-p_0)*[(x-p_1)*(x-p_2) + (x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_1)*[(x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_2)*(x-p_3)*(x-p_4) is two times a prime. The correspondence with A127491: the coefficient [x^2] of the polynomial (x-p_0)*(x-p_1)*..*(x-p_4) is the sum of 10 products of a set of 3 out of the 5 primes. Here the sum is restricted to the 6 products where the two largest of the 3 primes are consecutive. - R. J. Mathar, Apr 23 2023

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338, A127339, A127340, A127341, A127342, A127343, A127345, A127346, A127347, A127348, A127349, A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127489, A127490, A127491.

isA127492 := proc(k)
    local x,j ;
    (x-ithprime(k))* mul( x-ithprime(k+j),j=1..2)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+2))* mul( x-ithprime(k+j),j=3..4) ;
    p := abs(coeff(expand(%/2),x,0)) ;
    if type(p,'integer') then
        isprime(p) ;
    else
        false ;
    end if ;
end proc:
for k from 1 to 900 do
    if isA127492(k) then
        printf("%a,",k) ;
    end if ;
end do: # R. J. Mathar, Apr 23 2023

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2] + Prime[x] Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3] + Prime[x + 1] Prime[x + 3]Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])/2], AppendTo[a, x]], {x, 1, 1000}]; a
prQ[{a_,b_,c_,d_,e_}]:=PrimeQ[(a b c+a c d+a d e+b c d+b d e+c d e)/2]; PrimePi/@Select[ Partition[ Prime[Range[1000]],5,1],prQ][[;;,1]] (* Harvey P. Dale, Apr 21 2023 *)

Definition simplified by R. J. Mathar, Apr 23 2023

Edited by Jon E. Schoenfield, Jul 23 2023

A125270 Coefficient of x^2 in polynomial whose zeros are 5 consecutive primes starting with the n-th prime.

Original entry on oeis.org

1358, 3954, 10478, 22210, 43490, 78014, 129530, 206650, 324350, 466270, 621466, 853742, 1132130, 1436690, 1870850, 2388050, 2886370, 3440410, 4133410, 4904906, 5926654, 7195670, 8425430, 9792950, 11040910, 12098990, 13917898, 16097810

Offset: 1

Artur Jasinski, Jan 16 2007

nonn

Sums of all distinct products of 3 out of 5 consecutive primes, starting with the n-th prime; value of 3rd elementary symmetric function on the 5 consecutive primes.

Cf. A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338, A127339, A127340, A127341, A127342, A127343, A127345, A127346, A127347, A127348, A127349, A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127489.

a = {}; Do[AppendTo[a, (Prime[x] Prime[x + 1] Prime[x + 2] + Prime[x] Prime[x + 1] Prime[x + 3] + Prime[x] Prime[x + 1] Prime[x + 4] + Prime[x] Prime[x + 2] Prime[x + 3] + Prime[x] Prime[x + 2] Prime[x + 4] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2] Prime[x + 3] + Prime[x + 1] Prime[x + 2] Prime[x + 4] + Prime[x + 1] Prime[x + 3] Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])], {x, 1, 100}]; a
fcp[{p_,q_,r_,s_,t_}]:=p*q(r+s+t)+(p+q)r(s+t)+(p+q+r)s*t; fcp/@Partition[ Prime[ Range[40]],5,1] (* Harvey P. Dale, Sep 05 2014 *)

Let p = Prime(n), q = Prime(n+1), r = Prime(n+2), s = Prime(n+3) and t = Prime(n+4). Then a(n) = p q (r+s+t) + (p + q) r (s + t) + (p + q + r) s t.

Edited and corrected by Franklin T. Adams-Watters, Jan 23 2007

A339866 Primes that are simultaneously the sums of 11, 13, and 15 consecutive primes.

Original entry on oeis.org

8472193, 14084311, 16569827, 28358851, 33546551, 45993127, 91174081, 123593753, 186861293, 205286087, 224010023, 227568853, 310359607, 335497667, 423104119, 454320901, 482749429, 492404317, 558048187, 560997023, 566428813, 700508971, 707060359, 715731761, 735276379

Offset: 1

Zak Seidov, Apr 24 2021

nonn

Intersection of A127340, A127341, A161612.
The first case with 17 consecutive primes is a(219) = 8410721789. Are there more such terms?
a(10) = 205286087 is the sum of k consecutive primes not only for k = 11, 13, and 15, but also for k=1 (i.e., a(10) is a prime), k=9, and k=233. - Jon E. Schoenfield, Apr 24 2021

			Sum_{k=61746..61756} prime(k) = Sum_{k=52937..52949} prime(k) = Sum_{k=46425..46439} prime(k) = 8472193, so 8472193 is a term. - _Jon E. Schoenfield_, Apr 24 2021

Cf. A127340, A127341, A161612, A213914.

Module[{nn=4*10^6,prs,p11,p13,p15},prs=Prime[Range[nn]];p11=Total/@Partition[prs,11,1];p13=Total/@Partition[prs,13,1]; p15=Total/@ Partition[ prs,15,1];Select[Intersection[ p11,p13,p15],PrimeQ]] (* Harvey P. Dale, Aug 14 2023 *)

A341338 a(n) is the smallest prime that is simultaneously the sum of 2n-1, 2n+1 and 2n+3 consecutive primes.

Original entry on oeis.org

83, 311, 55813, 437357, 1219789, 8472193, 9496853, 6484103, 2166953, 37296143, 12671599, 13432571, 14968909, 145616561, 732092831, 220872569, 1381099933, 93482633, 4142423, 87030017, 3193060007, 736535783, 6390999871, 280886077, 464341303, 268231657, 686836817, 9000046663

Offset: 1

Zak Seidov, Apr 25 2021

nonn

			For n = 1: 83 = 23 + 29 + 31 = 11 + 13 + 17 + 19 + 23, and 83 is the smallest prime that is the sum of 1, 3 and 5 consecutive primes, so a(1) = 83.

Cf. A034962, A034965, A082246, A082251, A127340, A127341, A161612.

Array[(k=1;
While[(i=Select[Intersection@@((Total/@Subsequences[Prime@Range@Prime[k++],{#}])&/@{2#-1,2#+1,2#+3}),PrimeQ])=={}];First@i)&,4] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

cp's OEIS Frontend

A215991 Primes that are the sum of 25 consecutive primes.

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A070934 Smallest prime equal to the sum of 2n+1 consecutive primes.

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A127340 Primes that are the sum of 11 consecutive primes.

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A127346 Primes in A127345.

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A082244 Smallest odd prime that is the sum of 2n+1 consecutive primes.

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A342439 Let S(n,k) denote the set of primes < 10^n which are the sum of k consecutive primes, and let K = maximum k >= 2 such that S(n,k) is nonempty; then a(n) = max S(n,K).

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A127492 Indices m of primes such that Sum_{k=0..2, k

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