A001043 -id:A001043 - cp's OEIS frontend

A127333 Numbers that are the sum of 6 consecutive primes.

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

A118134 Primes p such that 4p is the sum of two consecutive primes.

Original entry on oeis.org

2, 3, 13, 17, 43, 67, 127, 137, 167, 193, 223, 283, 487, 563, 613, 617, 643, 647, 773, 1033, 1187, 1193, 1277, 1427, 1453, 1483, 1543, 1663, 1847, 1949, 2027, 2143, 2297, 2371, 2423, 2437, 2477, 2503, 2609, 2683, 2843, 2857, 2927, 3119, 3137, 3163, 3253, 3433

Offset: 1

Anton Vrba (antonvrba(AT)yahoo.com), May 13 2006

nonn

From Zak Seidov, Jun 18 2016: (Start)
Minimal difference between odd terms is 4.
a(n+1) - a(n) = 4 for n = {3, 15, 17, 147, 209, 277, 414, 422, 495, 825, 1053, 1380, 1504, 2078, 2264, 2375, 2605, 4224, 4495, 5180, 5825, 6497, 7107, 7372, 8951} and a(n) = {13, 613, 643, 16183, 24763, 37993, 63853, 65323, 81703, 154153, 210853, 295873, 327823, 479023, 537583, 568903, 632323, 1111723, 1195543, 1415833, 1626433, 1853443, 2060503, 2146813, 2702893} == 13 mod 30. (End)

			13 is there because it is prime and 4*13 = 23+29.

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

Cf. A001043 (sums of two consecutive primes).

pr = Prime[Range[1000]]; Select[(Total /@ Partition[pr, 2, 1])/4, PrimeQ] (* Zak Seidov, Jun 29 2017 *)

is(n)=isprime(n) && precprime(2*n)+nextprime(2*n)==4*n \\ Charles R Greathouse IV, Apr 24 2015

Edited by Don Reble, Jul 23 2006

A126148 Primes p such that pq+p+q is prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 23, 41, 43, 47, 59, 79, 83, 89, 101, 109, 113, 137, 163, 167, 173, 223, 229, 257, 311, 383, 389, 409, 419, 439, 443, 479, 521, 547, 557, 577, 593, 613, 643, 647, 683, 773, 797, 809, 811, 853, 953, 983, 1019, 1049, 1097, 1109, 1151, 1171

Offset: 1

J. M. Bergot, Mar 07 2007

			Take p = 13 and q = 17: product is 221 and sum is 30; add them to get 251, a prime. So 13 is a member.

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

Cf. A096342, A000040, A001043, A006094, A126199, A096342.

a:=proc(n) if isprime(ithprime(n)*ithprime(n+1) +ithprime(n) +ithprime(n+1)) then ithprime(n) fi end: seq(a(n), n=1..250); # Emeric Deutsch, Mar 08 2007

Prime@Select[Range[200], PrimeQ[Prime[ # ]Prime[ # + 1] + Prime[ # ] + Prime[ # + 1]] &] (* Ray Chandler, Mar 07 2007 *)

v=List();p=2;forprime(q=3,1e4, if(isprime(p*q+p+q), listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jul 26 2012

Extended by Ray Chandler, Emeric Deutsch and Robert G. Wilson v, Mar 07 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

A127338 Numbers that are the sum of 11 consecutive primes.

Original entry on oeis.org

160, 195, 233, 271, 311, 353, 399, 443, 491, 539, 583, 631, 677, 725, 779, 833, 883, 931, 979, 1025, 1081, 1139, 1197, 1253, 1313, 1367, 1423, 1483, 1543, 1607, 1673, 1727, 1787, 1843, 1901, 1951, 2011, 2077, 2141, 2203, 2263, 2323, 2383, 2443, 2507

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = absolute value of coefficient of x^10 of the polynomial Product_{j=0..10} (x - prime(n+j)) of degree 11; the roots of this polynomial are prime(n), ..., prime(n+10).

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

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127339, A127340.

[&+[ NthPrime(n+k): k in [0..10] ]: n in [1..70] ]; // Vincenzo Librandi, Apr 03 2011

f[n_] := Sum[Prime[n + i], {i, 0, 10}]; Array[f, 45]
Plus @@@ Partition[ Prime@ Range@ 55, 11, 1] (* Robert G. Wilson v, Jan 13 2011 *)

{m=45;k=11;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007

{m=45;k=11;for(n=1,m,print1(abs(polcoeff(prod(j=0,k-1,(x-prime(n+j))),k-1)),","))} \\ Klaus Brockhaus, Jan 13 2007

Edited by Klaus Brockhaus, Jan 13 2007

A127334 Numbers that are the sum of 7 consecutive primes.

Original entry on oeis.org

58, 75, 95, 119, 143, 169, 197, 223, 251, 281, 311, 341, 371, 401, 431, 463, 493, 523, 559, 593, 625, 659, 689, 719, 757, 791, 827, 863, 905, 947, 991, 1027, 1063, 1099, 1139, 1171, 1211, 1247, 1281, 1313, 1351, 1395, 1441, 1479, 1519, 1561, 1603, 1643

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = absolute value of coefficient of x^6 of the polynomial Product_{j=0..6} (x - prime(n+j)) of degree 7; the roots of this polynomial are prime(n), ..., prime(n+6).

Muniru A Asiru, Table of n, a(n) for n = 1..1000

Cf. A001043, A011974, A034961, A034963, A034964, A082246, A127333, A127335, A127336, A127337, A127338, A127339.

P:=Filtered([1..1000],IsPrime);; List([0..50],n->Sum([1+n..7+n],i->P[i])); # Muniru A Asiru, Apr 16 2018

[&+[ NthPrime(n+k): k in [0..6] ]: n in [1..70] ]; // Vincenzo Librandi, Apr 03 2011

seq(add(ithprime(i),i=n..6+n),n=1..50); # Muniru A Asiru, Apr 16 2018

a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 6}]], {x, 1, 50}]; a
Total/@Partition[Prime[Range[60]],7,1] (* Harvey P. Dale, May 14 2023 *)

{m=48;k=7;for(n=0,m-1,print1(a=sum(j=1,k,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007

{m=48;k=7;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(7))
print([a(i) for i in range(1, 50)]) # Indranil Ghosh, Mar 18 2017

BB = primes_first_n(62)
L = []
for i in range(55):
    L.append(sum(BB[i+j] for j in range(7)))
L
# Zerinvary Lajos, May 14 2007

Edited by Klaus Brockhaus, Jan 13 2007

A054268 Sum of composite numbers between prime p and nextprime(p) is a repdigit.

Original entry on oeis.org

3, 5, 109, 111111109, 259259257

Offset: 1

Patrick De Geest, Apr 15 2000

No additional terms below 472882027.

No additional terms below 10^58. - Chai Wah Wu, Jun 01 2024

			a(5) is ok since between 259259257 and nextprime 259259261 we get the sum 259259258 + 259259259 + 259259260 which yield repdigit 777777777.

Eric Weisstein's World of Mathematics, Repdigit

Cf. A010785, A028987, A028988, A046933, A054264, A054265, A054266, A054267.

repQ[n_]:=Count[DigitCount[n],0]==9; Select[Prime[Range[2,14500000]], repQ[Total[Range[#+1,NextPrime[#]-1]]]&] (* Harvey P. Dale, Jan 29 2011 *)

from sympy import prime
A054268 = [prime(n) for n in range(2,10**5) if len(set(str(int((prime(n+1)-prime(n)-1)*(prime(n+1)+prime(n))/2)))) == 1]
# Chai Wah Wu, Aug 12 2014

from itertools import count, islice
from sympy import isprime, nextprime
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A054268_gen(): # generator of terms
    for l in count(1):
        c = []
        for m in range(1,10):
            k = m*(10**l-1)//9<<1
            for a, b in diop_quadratic((x-y-1)*(x+y)-k):
                if isprime(b) and a == nextprime(b):
                    c.append(b)
        yield from sorted(c)
A054268_list = list(islice(A054268_gen(),5)) # Chai Wah Wu, Jun 01 2024

Numbers A000040(n) for n > 1 such that A001043(n)*(A001223(n)-1)/2 is in A010785. - Chai Wah Wu, Aug 12 2014

Offset changed by Andrew Howroyd, Aug 14 2024

A071216 a(n) is the largest prime factor of prime(n) + prime(n+1).

Original entry on oeis.org

5, 2, 3, 3, 3, 5, 3, 7, 13, 5, 17, 13, 7, 5, 5, 7, 5, 2, 23, 3, 19, 3, 43, 31, 11, 17, 7, 3, 37, 5, 43, 67, 23, 3, 5, 11, 5, 11, 17, 11, 5, 31, 3, 13, 11, 41, 31, 5, 19, 11, 59, 5, 41, 127, 13, 19, 5, 137, 31, 47, 3, 5, 103, 13, 7, 3, 167, 19, 29, 13, 89, 11, 37, 47, 127, 193, 131, 19

Offset: 1

Labos Elemer, May 17 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530, A001043, A071215.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]],2]; Table[pf[Prime[w+1]+Prime[w]], {w,1,128}]

a(n) = vecmax(factor(prime(n)+prime(n+1))[,1]); \\ Michel Marcus, Aug 29 2019

a(n) = A006530(A001043(n)).

A096379 a(n) = prime(n) + prime(n+1) - prime(n+2).

Original entry on oeis.org

0, 1, 1, 5, 7, 11, 13, 13, 21, 23, 27, 35, 37, 37, 41, 51, 53, 57, 65, 65, 69, 73, 75, 85, 95, 97, 101, 103, 95, 109, 121, 129, 127, 137, 143, 145, 153, 157, 161, 171, 169, 179, 187, 191, 185, 187, 207, 221, 223, 223, 231, 229, 235, 245, 251, 261, 263, 267, 275, 271

Offset: 1

Cino Hilliard, Aug 04 2004

Sequence is non-monotonic: see, e.g., a(29), a(33), and a(41). - Zak Seidov, Jan 21 2013

Ishikawa proved that a(n) > 0 for n > 1. - Jonathan Sondow, Feb 13 2014

			a(1) = prime(1) + prime(2) - prime(3) = 2 + 3 - 5 = 0.
a(25) = prime(25) + prime(26) - prime(27) = 97 + 101 - 103 = 95.

Michel Marcus, Table of n, a(n) for n = 1..5000 (terms 1..1000 from Zak Seidov)
Heihachiro Ishikawa, Über die Verteilung der Primzahlen, Sci. Rep. Tokyo Bunrika Daigaku, Sect. A 2 (1934), 27-40.

Cf. A000040, A001043, A001223.

[NthPrime(n)+NthPrime(n+1)-NthPrime(n+2):n in [1..60]]; //  Marius A. Burtea, Aug 17 2019

A096379 := proc(n)
    ithprime(n+1)+ithprime(n)-ithprime(n+2) ;
end proc:
seq(A096379(n),n=1..80) ; # R. J. Mathar, Sep 10 2016

#[[1]] + #[[2]] - #[[3]] & /@ Partition[Prime[Range[62]], 3, 1] (* Zak Seidov, Apr 09 2013 *)
ListConvolve[{-1,1,1},Prime[Range[100]]] (* Zak Seidov, Dec 03 2014 *)

g(n)=for(x=1,n,print1(prime(x)+prime(x+1)-prime(x+2)","))

first(n)=my(v=vector(n),p=2,q=3,k); forprime(r=5,, if(k++>n, break); v[k]=p+q-r; p=q; q=r); v \\ Charles R Greathouse IV, Oct 03 2017

a(n) = A001043(n) - A000040(n+2). - A.H.M. Smeets, Aug 17 2019

a(n) = A000040(n) - A001223(n+1). - Jon Maiga, Aug 17 2019

Edited by Zak Seidov, Aug 27 2012

Definition reworded by N. J. A. Sloane, Aug 27 2012

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A127333 Numbers that are the sum of 6 consecutive primes.

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A127335 Numbers that are the sum of 8 successive primes.

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A118134 Primes p such that 4p is the sum of two consecutive primes.

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A126148 Primes p such that pq+p+q is prime, where q is the next prime after p.

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A127336 Numbers that are the sum of 9 consecutive primes.

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A127338 Numbers that are the sum of 11 consecutive primes.

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