A034962 -id:A034962 - cp's OEIS frontend

A207527 Primes that are the sum of three consecutive primes in A034962.

131, 251, 337, 503, 743, 929, 1447, 1597, 3023, 3919, 4051, 4157, 5087, 5897, 6133, 7649, 7877, 8269, 9181, 9433, 9721, 11411, 11677, 13151, 13757, 14009, 14533, 18133, 18493, 18743, 20563, 21023, 21247, 22651, 22921, 23057, 23801, 23893, 24359, 24733, 24809

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 18 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A034962.

t = Select[Table[Prime[n] + Prime[n + 1] + Prime[n + 2], {n, 1000}], PrimeQ]; Select[Table[t[[n]] + t[[n + 1]] + t[[n + 2]], {n, Length[t] - 2}], PrimeQ]

A117714 a(n) = (A034962(n) - A152470(n))/2.

Original entry on oeis.org

6, 9, 12, 18, 21, 26, 30, 34, 42, 56, 64, 69, 72, 81, 86, 102, 111, 144, 150, 160, 165, 198, 217, 231, 274, 282, 288, 300, 312, 342, 348, 351, 381, 393, 405, 414, 432, 453, 459, 465, 473, 495, 501, 515

Offset: 1

Roger L. Bagula, Apr 13 2006

The sequence is always increasing.

Cf. A072225, A073681.

a = Flatten[Table[If[PrimeQ[Prime[n] + Prime[n + 1] + Prime[n + 2]] == True, If [Prime[n] + Prime[n + 1] + Prime[n + 2] - Prime[m] == 0, {( Prime[m] - Prime[n + 2])/2}, {}], {}], {n, 1, 100}, {m, 1, 500}]]

Description simplified by the Assoc. Eds. of the OEIS, Jun 27 2010

A034961 Sums of three consecutive primes.

Original entry on oeis.org

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

A215991 Primes that are the sum of 25 consecutive primes.

Original entry on oeis.org

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

A073681 Smallest of three consecutive primes whose sum is a prime.

Original entry on oeis.org

5, 7, 11, 17, 19, 23, 29, 31, 41, 53, 61, 67, 71, 79, 83, 101, 109, 139, 149, 157, 163, 197, 211, 229, 271, 281, 283, 293, 311, 337, 347, 349, 379, 389, 401, 409, 431, 449, 457, 463, 467, 491, 499, 509, 547, 617, 641, 653, 659, 661, 701, 719, 743, 751, 757

Offset: 1

Amarnath Murthy, Aug 11 2002

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harry J. Smith)

Cf. A152469, A152470, A174742, A034962, A152468, A180948, A189571, A180950, A226380.

[NthPrime(n): n in [0..200] | IsPrime(NthPrime(n)+NthPrime(n+1)+ NthPrime(n+2))]; // Vincenzo Librandi, May 06 2015

t0:=[];
t1:=[];
t2:=[];
for i from 1 to 1000 do
t3:=ithprime(i)+ithprime(i+1)+ithprime(i+2);
if isprime(t3) then
t0:=[op(t0),i];
t1:=[op(t1),ithprime(i)];
t2:=[op(t2),ithprime(i+2)];
fi;
od:
t1;

Transpose[Select[Partition[Prime[Range[200]],3,1],PrimeQ[Total[#]]&]] [[1]] (* Harvey P. Dale, Jan 25 2012 *)

forprime(p=1,1000, pp=nextprime(p+1); if(isprime(p+pp+nextprime(pp+1)),print1(p",")))

A073681(n,print_all=0,start=3)={my(r,q=1);forprime(p=start,, isprime(r+(r=q)+(q=p)) & (n-- ||return(precprime(r-1))) & print_all & print1(precprime(r-1)","))} \\ M. F. Hasler, Dec 18 2012

Conjecture: for n -> oo, a(n) ~ prime(n) * (log(prime(n)))^C, where C = 8/Pi^2 (cf. A217739). - Alain Rocchelli, Sep 04 2023

More terms from Ralf Stephan, Mar 20 2003

More cross-references from Harvey P. Dale, Jun 05 2013

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

A127341 Primes that can be written as the sum of 13 consecutive primes.

Original entry on oeis.org

691, 863, 983, 1153, 1283, 1553, 1621, 1753, 1823, 2111, 2239, 2311, 2963, 3191, 3617, 3853, 4099, 4357, 4519, 4597, 4999, 5081, 5393, 5471, 5623, 5693, 5849, 6229, 6491, 6971, 7349, 7673, 8123, 8191, 8669, 8933, 9391, 10141, 10499, 10949, 11273

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

			691 = prime(10) + prime(11) + ... + prime(22) = 29 + 31 + ... + 79.

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

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

Select[Table[Sum[Prime[i], {i, n, n + 12}], {n, 1, 150}], PrimeQ[ # ] &]
Select[Total/@Partition[Prime[Range[200]],13,1],PrimeQ] (* Harvey P. Dale, Aug 13 2021 *)

Edited by Stefan Steinerberger, Jul 31 2007

A072225 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 18, 19, 20, 22, 23, 26, 29, 34, 35, 37, 38, 45, 47, 50, 58, 60, 61, 62, 64, 68, 69, 70, 75, 77, 79, 80, 83, 87, 88, 90, 91, 94, 95, 97, 101, 113, 116, 119, 120, 121, 126, 128, 132, 133, 134

Offset: 1

Joseph L. Pe, Jul 04 2002

The sequence contains 298884 terms <= 2000000, so nearly 15% of the numbers <= 2000000 are in the sequence. - Dmitry Kamenetsky, Aug 08 2015
Heuristically, we might expect the "probability" of n being in the sequence to be on the order of 1/log(n). - Robert Israel, Aug 09 2015
The first 8 consecutive integers in this sequence start from 8744076. - Dmitry Kamenetsky, Aug 28 2015
The first 9 consecutive integers in this sequence start from 697642916. - Dmitry Kamenetsky, Sep 08 2015
Vladimir Chirkov found that the first 10 and 11 consecutive integers in this sequence start from 23169509240 and 29165083170, respectively (see The Prime Puzzles link and A386275). - Dmitry Kamenetsky, Sep 08 2015

			9 is in the sequence because prime(9) + prime(10) + prime(11) = 23 + 29 + 31 = 83 is a prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 798. A nice puzzle by Kamenetski, The Prime Puzzles & Problems Connection.

Cf. A034962, A073681, A117673, A152469, A152470.

Cf. A386275 (first occurrences of runs).

[n: n in [0..600]| IsPrime(NthPrime(n)+NthPrime(n+1)+NthPrime(n+2))]; // Vincenzo Librandi, Apr 06 2011

a:=proc(n) if isprime(ithprime(n)+ithprime(n+1)+ithprime(n+2))=true then n else fi end: seq(a(n),n=1..150); # Emeric Deutsch, Apr 24 2006

Select[Range[10^4], PrimeQ[Prime[ # ] + Prime[ # + 1] + Prime[ # + 2]] &]

isok(n)=isprime(prime(n)+prime(n+1)+prime(n+2)) \\ Anders Hellström, Aug 20 2015

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

cp's OEIS Frontend

A207527 Primes that are the sum of three consecutive primes in A034962.

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A117714 a(n) = (A034962(n) - A152470(n))/2.

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A034961 Sums of three consecutive primes.

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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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A073681 Smallest of three consecutive primes whose sum is a prime.

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A127340 Primes that are the sum of 11 consecutive primes.

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