A011974 -id:A011974 - cp's OEIS frontend

A099655 a[n]=A098085[n]-A096215[n], difference between next and previous primes to A011974[n], the sum of two consecutive primes.

4, 4, 2, 2, 6, 2, 6, 2, 6, 2, 4, 6, 6, 8, 4, 4, 14, 4, 2, 10, 6, 6, 6, 10, 2, 12, 12, 12, 12, 2, 6, 6, 6, 10, 14, 4, 14, 14, 10, 4, 8, 6, 6, 8, 8, 10, 6, 8, 8, 2, 12, 8, 8, 6, 12, 18, 18, 10, 6, 6, 6, 2, 2, 12, 12, 6, 12, 8, 10, 8, 10, 8, 4, 6, 8, 4, 14, 12, 2, 2, 14, 14, 14, 14, 2, 20, 20, 8, 10, 8

Offset: 1

Labos Elemer, Nov 17 2004

nonn

			n=8, p(8)+p(9)=19+23=42,a[8]=43-41=2=a(8).

Cf. A098085, A096215, A011974.

Mathematica
```
<Harvey P. Dale, Mar 07 2017 *)
```

a(n)=NextPrime[p(n)+p(n+1)]-PreviousPrime[p(n)+p(n+1)]

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

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

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

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

A127337 Numbers that are the sum of 10 consecutive primes.

Original entry on oeis.org

129, 158, 192, 228, 264, 300, 340, 382, 424, 468, 510, 552, 594, 636, 682, 732, 780, 824, 870, 912, 954, 1008, 1060, 1114, 1164, 1216, 1266, 1320, 1376, 1434, 1494, 1546, 1596, 1650, 1704, 1752, 1800, 1854, 1914, 1974, 2030, 2084, 2142, 2192, 2250, 2310, 2374

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) is the absolute value of coefficient of x^9 of the polynomial Product_{j=0..9} (x - prime(n+j)) of degree 10; the roots of this polynomial are prime(n), ..., prime(n+9).

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

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127338, A127339.

[&+[ NthPrime(n+k): k in [0..9] ]: n in [1..90] ]; // Vincenzo Librandi, Apr 03 2011

A127337 := proc(n)
    local i ;
    add(ithprime(n+i),i=0..9) ;
end proc:
seq(A127337(n),n=1..30) ; # R. J. Mathar, Apr 24 2023

a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 9}]], {x, 1, 50}]; a
Table[Plus@@Prime[Range[n, n + 9]], {n, 50}] (* Alonso del Arte, Feb 15 2011 *)
ListConvolve[ConstantArray[1, 10], Prime[Range[50]]]
Total/@Partition[Prime[Range[60]],10,1] (* Harvey P. Dale, Jan 31 2013 *)

{m=46;k=10;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007

{m=46;k=10;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

from sympy import prime
def a(n): return sum(prime(n + i) for i in range(10))
print([a(n) for n in range(1, 48)]) # Michael S. Branicky, Dec 09 2021

# faster version for generating initial segment of sequence
from sympy import nextprime
def aupton(terms):
    alst, plst = [], [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
    for n in range(terms):
        alst.append(sum(plst))
        plst = plst[1:] + [nextprime(plst[-1])]
    return alst
print(aupton(47)) # Michael S. Branicky, Dec 09 2021

a(n) = A127336(n)+A000040(n+9). - R. J. Mathar, Apr 24 2023

Edited by Klaus Brockhaus, Jan 13 2007

A127333 Numbers that are the sum of 6 consecutive primes.

Original entry on oeis.org

41, 56, 72, 90, 112, 132, 156, 180, 204, 228, 252, 280, 304, 330, 358, 384, 410, 434, 462, 492, 522, 552, 580, 606, 630, 660, 690, 724, 756, 796, 834, 864, 896, 926, 960, 990, 1020, 1054, 1084, 1114, 1140, 1172, 1214, 1250, 1286, 1322, 1362, 1392, 1420

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) is the absolute value of coefficient of x^5 of the polynomial Prod_{j=0,5}(x-prime(n+j)) of degree 6; the zeros of this polynomial are prime(n), ..., prime(n+5).

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

Cf. A011974, A001043, A034961, A034963, A034964, A127334, A127335, A127336, A127337, A127338, A127339.

[&+[ NthPrime(n+k): k in [0..5] ]: n in [1..80] ]; /* Vincenzo Librandi, Apr 03 2011 */

a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 5}]], {x, 1, 50}]; a
Total/@Partition[Prime[Range[60]],6,1] (* Harvey P. Dale, Mar 12 2015 *)

{m=50;k=6;for(n=0,m-1,print1(a=sum(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 12 2007

{m=50;k=6;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 12 2007

Edited by Klaus Brockhaus, Jan 12 2007

A127335 Numbers that are the sum of 8 successive primes.

Original entry on oeis.org

77, 98, 124, 150, 180, 210, 240, 270, 304, 340, 372, 408, 442, 474, 510, 546, 582, 620, 660, 696, 732, 768, 802, 846, 888, 928, 966, 1012, 1056, 1104, 1154, 1194, 1236, 1278, 1320, 1362, 1404, 1444, 1480, 1524, 1574, 1622, 1670, 1712, 1758, 1802, 1854, 1900

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) is the absolute value of coefficient of x^7 of the polynomial Prod_{j=0,7}(x-prime(n+j)) of degree 8; the roots of this polynomial are prime(n), ..., prime(n+7).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127336, A127337, A127338, A127339.

[&+[ NthPrime(n+k): k in [0..7] ]: n in [1..90] ]; // Vincenzo Librandi, Apr 03 2011

S:= [0,op(ListTools:-PartialSums(select(isprime, [2,seq(i,i=3..1000,2)])))]:
S[9..-1]-S[1..-9]; # Robert Israel, Nov 27 2017

a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 7}]], {x, 1, 50}]; a
Total/@Partition[Prime[Range[60]],8,1] (* Harvey P. Dale, Sep 10 2019 *)

{m=48;k=8;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007

{m=48;k=8;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

a(n)=my(p=prime(n));p+sum(i=2,8,p=nextprime(p+1)) \\ Charles R Greathouse IV, Apr 19 2015

a(n) ~ 8n log n. - Charles R Greathouse IV, Apr 19 2015

Edited by Klaus Brockhaus, Jan 13 2007

A127336 Numbers that are the sum of 9 consecutive primes.

Original entry on oeis.org

100, 127, 155, 187, 221, 253, 287, 323, 363, 401, 439, 479, 515, 553, 593, 635, 679, 721, 763, 803, 841, 881, 929, 977, 1025, 1067, 1115, 1163, 1213, 1267, 1321, 1367, 1415, 1459, 1511, 1555, 1601, 1643, 1691, 1747, 1801, 1851, 1903, 1951, 1999, 2053

Offset: 1

Artur Jasinski, Jan 11 2007

a(n) = absolute value of coefficient of x^8 of the polynomial Product_{j=0..8}(x - prime(n+j)) of degree 9; the roots of this polynomial are prime(n), ..., prime(n+8).

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127337, A127338, A127339, A082251, A228200.

[&+[ NthPrime(n+k): k in [0..8] ]: n in [1..100] ]; // Vincenzo Librandi, Apr 03 2011

A127336 = {}; Do[AppendTo[A127336, Sum[Prime[x + n], {n, 0, 8}]], {x, 1, 50}]; A127336 (* Artur Jasinski, Jan 11 2007 *)
Table[Plus@@Prime[Range[n, n + 8]], {n, 50}] (* Alonso del Arte, Aug 27 2013 *)
Total/@Partition[Prime[Range[60]],9,1] (* Harvey P. Dale, Nov 18 2020 *)

{m=46;k=9;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007
{m=46;k=9;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

from sympy import prime
def a(x): return sum([prime(x + n) for n in range(9)])
print([a(i) for i in range(1, 50)]) # Indranil Ghosh, Mar 18 2017

a(n) = A127335(n)+A000040(n+8). - R. J. Mathar, Apr 24 2023

Edited by Klaus Brockhaus, Jan 13 2007

cp's OEIS Frontend

A099655 a[n]=A098085[n]-A096215[n], difference between next and previous primes to A011974[n], the sum of two consecutive primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A034961 Sums of three consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Sage

Formula

A034963 Sums of four consecutive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A034962 Primes that are the sum of three consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Python

A034964 Sums of five consecutive primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A034965 Primes that are sum of five consecutive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs