A034962 -id:A034962 - cp's OEIS frontend

A060328 Primes which are the sum of three consecutive composite numbers.

23, 31, 41, 59, 67, 71, 109, 113, 131, 139, 157, 199, 211, 239, 251, 269, 293, 311, 337, 379, 383, 409, 419, 487, 491, 499, 503, 521, 571, 599, 631, 701, 751, 769, 773, 787, 829, 877, 881, 919, 941, 953, 991, 1009, 1013, 1039, 1049, 1061, 1103, 1117, 1151

Offset: 1

Robert G. Wilson v, Mar 30 2001

nonn

"Consecutive" necessarily means consecutive in the list of composite numbers as opposed to consecutive in the integers, as the sum of any 3 consecutive integers is a multiple of 3. - Peter Munn, Aug 20 2023

			a(3) = 41 is equal to 12+14+15.

Primes that are the sum of other numbers of consecutive composite numbers: A060254 (2), A060329 (4), A060330 (5), A060331 (6), A060332 (7), A060333 (8). See also A037174.
Cf. A034962.
Complement within A166039\{5, 11} of A151741.

composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); b = {}; Do[ p = composite[ n ] + composite[ n + 1 ] + composite[ n + 2 ]; If[ PrimeQ[ p ], b = Append[ b, p ] ], {n, 1, 1000} ]; b

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

A065867 Primes which are the sum of a prime number of consecutive primes.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Dec 07 2001

			5 = 2 + 3.
23 = 5 + 7 + 11.
31 = 7 + 11 + 13.
41 = 11 + 13 + 17.
53 = 5 + 7 + 11 + 13 + 17.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001043, A034707, A034962, A034965, A138591, A348790, A348791.

Cf. A034962 (subsequence).

lst={};Do[s=Prime[m];k=1;Do[p=Prime[n];s+=p;k++;If[PrimeQ[s]&&PrimeQ[k],If[s<=10837,AppendTo[lst,s]]],{n,m+1,5*5!}],{m,5*5!}];lst=Take[Union@lst,500] (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)
Module[{nn=60,prs},prs=Prime[Range[nn]];Take[Select[Union[ Flatten[ Table[ Total/@ Partition[prs,n,1],{n,prs}]]],PrimeQ],nn]] (* Harvey P. Dale, Aug 12 2016 *)

A164129 Primes that are the sums of cubes of three consecutive primes.

Original entry on oeis.org

66347, 199081, 332207, 581237, 733123, 1047691, 2647943, 3612799, 7505063, 10620793, 22715029, 32180581, 36355409, 60621553, 76753387, 98784001, 116319367, 147594259, 162516943, 177616529, 216596449, 252725563, 343774313

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

nonn

			23^3+29^3+31^3=66347, 37^3+41^3+43^3=199081, 43^3+47^3+53^3=332207,..

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A133530, A034962

lst={};Do[p=Prime[n]^3+Prime[n+1]^3+Prime[n+2]^3;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst
Select[Total[#^3]&/@Partition[Prime[Range[100]],3,1],PrimeQ] (* Harvey P. Dale, Nov 04 2015 *)

Edited by Charles R Greathouse IV, Oct 12 2009

A174518 Sums of two consecutive primes and composite numbers in-between.

Original entry on oeis.org

5, 12, 18, 45, 36, 75, 54, 105, 182, 90, 238, 195, 126, 225, 350, 392, 180, 448, 345, 216, 532, 405, 602, 837, 495, 306, 525, 324, 555, 1800, 645, 938, 414, 1584, 450, 1078, 1120, 825, 1190, 1232, 540, 2046, 576, 975, 594, 2665, 2821, 1125, 684, 1155, 1652

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 21 2010

nonn

Cf. A034962, A054265.

f[n_,x_]:=n*x+x*(x+1)/2;Table[Prime[n]+f[Prime[n],Prime[n+1]-Prime[n]-1]+Prime[n+1],{n,5!}]

a(n) = Sum_{k=prime(n)..prime(n+1)} k. - Wesley Ivan Hurt, Apr 27 2021

A211170 Primes that are sum of both three and five consecutive primes.

Original entry on oeis.org

83, 199, 311, 941, 1151, 1381, 2357, 3121, 4337, 4363, 4957, 5059, 7039, 8069, 8117, 8161, 8389, 8627, 8819, 8971, 9011, 9349, 10211, 10253, 13127, 14813, 16249, 19207, 19717, 21377, 23143, 24329, 32983, 34807, 38113, 39623, 41141, 44279, 45061, 45979, 58403

Offset: 1

Zak Seidov, Jan 31 2013

nonn

Intersection of A034962 and A034965.

			a(1) = 83 = A034962(6) = 23 + 29 + 31 = A034965(3) = 11 + 13 + 17 + 19 + 23.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A034962, A034965, A073681, A152468.

Module[{prs=Prime[Range[3000]],pr3,pr5},pr3=Select[Total/@Partition[ prs, 3, 1], PrimeQ];pr5=Select[Total/@Partition[prs,5,1],PrimeQ];Intersection[ pr3,pr5]] (* Harvey P. Dale, Oct 24 2016 *)

A259772 Primes p such that p^3 + q^2 + r is also prime, where p,q,r are consecutive primes.

Original entry on oeis.org

3, 17, 19, 43, 53, 89, 107, 149, 293, 401, 439, 449, 659, 809, 821, 937, 1009, 1031, 1091, 1097, 1123, 1163, 1181, 1259, 1277, 1367, 1427, 1657, 1721, 1777, 1789, 1811, 1987, 2027, 2063, 2207, 2333, 2417, 2503, 2657, 2713, 3067, 3079, 3083, 3251, 3389, 3491, 3527

Offset: 1

K. D. Bajpai, Jul 05 2015

nonn

			a(2) = 17 is prime: 17^3 + 19^2 + 23 = 5297 which is also prime.
a(3) = 19 is prime: 19^3 + 23^2 + 29 = 7417 which is also prime.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A034962, A133529, A133530, A258269, A304292.

[p: p in PrimesUpTo (3000) | IsPrime(k) where k is (p^3 + NextPrime(p)^2 + NextPrime(NextPrime(p)))];

select(n -> isprime(n) and isprime((n)^3+nextprime(n)^2+nextprime(nextprime((n)))), [seq(n, n=1..10000)]);

Select[Prime[Range[1000]], PrimeQ[#^3 + NextPrime[#]^2 + NextPrime[NextPrime[#]]]&]
Select[Partition[Prime[Range[500]],3,1],PrimeQ[#[[1]]^3+ #[[2]]^2+ #[[3]]]&][[All,1]] (* Harvey P. Dale, Dec 23 2021 *)

forprime(p=1, 3000, q=nextprime(p+1); r=nextprime(q+1); k=(p^3 + q^2 + r); if(isprime(k), print1(p,", ")))

A050207 Primes of the form p + (smallest prime >= p+1) + (smallest prime >= p+3) where p is a prime.

Original entry on oeis.org

23, 29, 41, 47, 59, 97, 131, 137, 223, 283, 311, 317, 367, 389, 457, 563, 587, 607, 677, 743, 839, 857, 907, 929, 941, 947, 1031, 1049, 1093, 1283, 1303, 1453, 1489, 1847, 1867, 1913, 1931, 1993, 2027, 2347, 2381, 2441, 2477, 2579, 2617, 2657

Offset: 1

Cino Hilliard, May 08 2003

nonn

Originally an erroneous version of A034962.

			p = 5: 23 = 5 + 7 + 11;
p = 7: 29 = 7 + 11 + 11;
p = 11: 41 = 11 + 13 + 17;
p = 13: 47 = 13 + 17 + 17;
p = 17: 59 = 17 + 19 + 23;
p = 19: 65 = 19 + 23 + 23 is not prime, so not in the sequence;
p = 23: 81 = 23 + 29 + 29 is not prime, so not in the sequence;
p = 29: 97 = 29 + 31 + 37.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A034962.

Select[Table[p+NextPrime[p]+NextPrime[p+2],{p,Prime[Range[300]]}],PrimeQ] (* Harvey P. Dale, Dec 04 2017 *)

psumpr3(n) = { c1=0; c2=0; forprime(x=3, n, s = nextprime(x)+nextprime(x+1)+nextprime(x+3); c1++; if(isprime(s), c2++; print1(s" ")); ); print(); print(c2/c1+.0) }

Edited by Joshua Zucker, Jan 27 2007

Definition clarified by Harvey P. Dale, Dec 04 2017

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

cp's OEIS Frontend

A060328 Primes which are the sum of three consecutive composite numbers.

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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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A065867 Primes which are the sum of a prime number of consecutive primes.

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A164129 Primes that are the sums of cubes of three consecutive primes.

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A174518 Sums of two consecutive primes and composite numbers in-between.

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A211170 Primes that are sum of both three and five consecutive primes.

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A259772 Primes p such that p^3 + q^2 + r is also prime, where p,q,r are consecutive primes.

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