A001043 -id:A001043 - cp's OEIS frontend

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

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 *)

A096278 Sums of successive sums of successive sums of successive primes.

Original entry on oeis.org

33, 50, 72, 96, 120, 144, 172, 206, 240, 274, 308, 336, 364, 402, 444, 480, 514, 548, 578, 610, 648, 692, 742, 786, 816, 840, 864, 900, 960, 1024, 1070, 1108, 1152, 1196, 1236, 1278, 1320, 1362, 1404, 1444, 1488, 1530, 1560, 1592, 1650, 1728, 1790, 1824

Offset: 1

Cino Hilliard, Jun 22 2004

If we consider the m-fold iterated "take sums of successive terms" operation acting on the primes, then for all m >= 1, the first term is always odd (and the only odd term); it is prime for m=1, 2, 4, 8, 21, 24, 27, 31, 40, 98,..., but not for m=3 (the present sequence). [Edited by M. F. Hasler, Jun 02 2017]

			The first two terms of SS order 1 is 13 and 20. 13+20 = 33 the first term of the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A096277, A001043.

Ss:= L -> L[1..-2]+L[2..-1]:
(Ss@@3)([seq(ithprime(i),i=1..100)]); # Robert Israel, Dec 28 2022

Nest[ListConvolve[{1,1},#]&,Prime[Range[100]],3] (* Paolo Xausa, Oct 31 2023 *)

f(n) = return(prime(n)+prime(n+1))
f1(n) = return(f(n)+f(n+1))
f2(n) = return(f1(n)+f1(n+1))
g(n) = for(x=1,n,print1(f2(x)","))

A096278(n,m=3)=for(k=0,m,prime(n+k)*binomial(m,k)) \\ or, to get a list:
A096278_vec(Nmax,m=3,v=primes(Nmax+m))=sum(k=0,m,binomial(m,k)*v[1+k,k-1-m]) \\ Alternatively, do m times v=v[^1]+v[^-1]. - M. F. Hasler, Jun 02 2017

Let f(n) = prime(n) + prime(n+1) f1(n) = f(n)+f(n+1) : SS of order 1 Then f2(n) = f1(n)+f1(n) : SS of order 2 is the general term of this sequence.
a(n) = A096277(n) + A096277(n+1). - M. F. Hasler, Jun 02 2017
a(n) = prime(n)+3*prime(n+1)+3*prime(n+2)+prime(n+3). - Robert Israel, Dec 28 2022

A098037 Number of prime divisors, counted with multiplicity, of the sum of two consecutive primes.

Original entry on oeis.org

1, 3, 3, 3, 4, 3, 4, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 7, 3, 6, 4, 5, 3, 3, 4, 4, 4, 6, 3, 6, 3, 3, 4, 7, 5, 4, 7, 4, 4, 6, 6, 4, 8, 4, 5, 3, 3, 5, 5, 4, 4, 7, 4, 3, 5, 4, 6, 3, 4, 4, 8, 6, 3, 6, 5, 7, 3, 5, 5, 5, 4, 4, 4, 5, 3, 3, 3, 4, 6, 5, 6, 4, 8, 4, 5, 3, 3, 5, 5, 4, 3, 4, 3, 5, 3, 4, 3, 5, 5, 7, 6, 7, 3, 5, 4

Offset: 1

Cino Hilliard, Sep 10 2004

Clearly sum of two consecutive primes prime(x) and prime(x+1) has more than 2 prime divisors for all x > 1.

			Prime(2) + prime(3) = 2*2*2, 3 factors, the second term in the sequence.

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

Cf. A071215, A251600 (greedy inverse).

A098037 := proc(n)
    ithprime(n)+ithprime(n+1) ;
    numtheory[bigomega](%) ;
end proc:
seq(A098037(n),n=1..40) ; # R. J. Mathar, Jan 20 2025

PrimeOmega[Total[#]]&/@Partition[Prime[Range[110]],2,1] (* Harvey P. Dale, Jun 14 2011 *)

b(n) = for(x=1,n,y1=(prime(x)+prime(x+1));print1(bigomega(y1)","))

a(n) = A001222(A001043(n)). - Michel Marcus, Feb 15 2014

Definition corrected by Andrew S. Plewe, Apr 08 2007

A226524 Cubes which are the sum of two consecutive primes.

Original entry on oeis.org

8, 216, 21952, 74088, 373248, 4251528, 5268024, 10648000, 10941048, 12812904, 14886936, 16003008, 25934336, 40707584, 54872000, 59319000, 114791256, 132651000, 199176704, 209584584, 259694072, 269586136, 306182024, 345948408, 373248000, 543338496, 567663552

Offset: 1

K. D. Bajpai, Aug 31 2013

			a(2) = 216: prime(28) + prime(29) = 107 + 109 = 216 = 6^3.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cubes in A001043.

Cf. A062703 (analog for squares), A061308 (lesser of the consecutive primes), A071220 (index of that prime), A074925 (a(n)^(1/3)).

KD: = proc() local a,b,c;  a: = ithprime(n) + ithprime(n+1); b:= evalf(a^(1/3)); if b=floor(b) then RETURN(a):  fi; end: seq(KD(), n=1..1000000);

Select[Total/@Partition[Prime[Range[155*10^5]],2,1],IntegerQ[Surd[#,3]]&] (* or *) stcpQ[n_]:=Module[{p1=NextPrime[Floor[n/2],-1],p2=NextPrime[Ceiling[n/2]]},n==p1+p2]; Select[Range[850]^3,stcpQ] (* The second program is much more efficient than the first. *) (* Harvey P. Dale, May 15 2022 *)

n=0; forstep(j=2, 55778, 2, c=j^3; c2=c/2; if(precprime(c2)+nextprime(c2)==c, n++; write("b226524.txt", n " " c))) /* Donovan Johnson, Sep 02 2013 */

A226524(n)=A074925(n)^3 \\ M. F. Hasler, Jan 03 2020

from itertools import count, islice
from sympy import nextprime, prevprime
def agen(): yield from (c for c in (k**3 for k in count(2, step=2)) if prevprime(c//2+1) + nextprime(c//2-1) == c)
print(list(islice(agen(), 27))) # Michael S. Branicky, May 24 2022

a(n) = A074925(n)^3 = A000040(i) + A000040(i+1) with i = A071220(n) = A000720(A061308(n)). - M. F. Hasler, Jan 03 2020

Edited by M. F. Hasler, Jan 03 2020

A298463 The first of two consecutive pentagonal numbers the sum of which is equal to the sum of two consecutive primes.

Original entry on oeis.org

70, 3577, 10795, 36895, 55777, 70525, 78547, 125137, 178365, 208507, 258130, 329707, 349692, 394497, 438751, 468442, 478555, 499105, 619852, 663005, 753667, 827702, 877455, 900550, 1025480, 1085876, 1169092, 1201090, 1211852, 1233520, 1339065, 1508512

Offset: 1

Colin Barker, Jan 19 2018

nonn

			70 is in the sequence because 70+92 (consecutive pentagonal numbers) = 162 = 79+83 (consecutive primes).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000040, A000326, A001043, A061275, A298462, A298464, A298465, A298466.

Select[Partition[PolygonalNumber[5,Range[1500]],2,1],CompositeQ[Total[#]/2]&&Total[#] == NextPrime[ Total[#]/2]+NextPrime[Total[#]/2,-1]&][[;;,1]] (* Harvey P. Dale, Jan 20 2024 *)

L=List(); forprime(p=2, 1600000, q=nextprime(p+1); t=p+q; if(issquare(12*t-8, &sq) && (sq-2)%6==0, u=(sq-2)\6; listput(L, (3*u^2-u)/2))); Vec(L)

from _future_ import division
from sympy import prevprime, nextprime
A298463_list, n, m = [], 1 ,6
while len(A298463_list) < 10000:
    k = prevprime(m//2)
    if k + nextprime(k) == m:
        A298463_list.append(n*(3*n-1)//2)
    n += 1
    m += 6*n-1 # Chai Wah Wu, Jan 20 2018

A092163 a(n) = prime(2n) + prime(2n+1).

Original entry on oeis.org

8, 18, 30, 42, 60, 78, 90, 112, 128, 144, 162, 186, 204, 216, 240, 268, 288, 308, 330, 352, 372, 390, 410, 450, 462, 480, 508, 532, 548, 564, 600, 624, 648, 684, 702, 726, 752, 772, 798, 828, 852, 872, 892, 918, 930, 966, 990, 1012, 1044, 1088, 1120, 1140

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 31 2004

nonn

			a(1) = 8 because p(2)= 3 and p(3) = 5.
a(2) = 18 because p(4)= 7 and p(5) = 11.
a(3) = 30 because p(6)= 13 and p(7) = 17.

Cf. A000040, A001043, A058296, A100479.

A092163 := proc(n)
    ithprime(2*n)+ithprime(2*n+1) ;
end proc:
seq(A092163(n),n=1..50) ; # R. J. Mathar, Jan 20 2025

Table[ Prime[2n] + Prime[2n + 1], {n, 50}] (* Robert G. Wilson v, Apr 22 2004 *)

a(n) = A001043(2*n). - R. J. Mathar, Apr 20 2009

a(n) = 2*A058296(n+1). - Hugo Pfoertner, Aug 11 2025

More terms from Robert G. Wilson v, Apr 22 2004

A096279 Sums of successive sums of successive sums of successive sums of successive primes.

Original entry on oeis.org

83, 122, 168, 216, 264, 316, 378, 446, 514, 582, 644, 700, 766, 846, 924, 994, 1062, 1126, 1188, 1258, 1340, 1434, 1528, 1602, 1656, 1704, 1764, 1860, 1984, 2094, 2178, 2260, 2348, 2432, 2514, 2598, 2682, 2766, 2848, 2932, 3018, 3090, 3152, 3242, 3378

Offset: 1

Cino Hilliard, Jun 22 2004

The first term is always odd and may be prime. The first terms of more and more successions produce A007443.

			2 3 5 7 11 successive primes
5 8 12 18 sums of successive primes
13 20 30 sums of successive sums of successive primes
33 50 sums of successive sums of successive sums of successive primes
83 sums of successive sums of successive sums of successive sums of successive primes

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A096277, A096278.

Nest[ListConvolve[{1,1},#]&,Prime[Range[100]],4] (* Paolo Xausa, Oct 31 2023 *)

\\ Iterated successive sums of successive sums... of successive primes.
\\ Input number of terms n and the order m. m=0 yields the primes.
sucsums(n, m) = { my(a, b, i, j, k); a = primes(n+m); b = vector(#a); for(i=1, m, for(j=1, n+m-i, b[j] = a[j]+ a[j+1]; ); a=b; ); for(k=1, n, print1(a[k], ", "); ) }

A098764 a(n) = 3p - q where p and q are consecutive primes.

Original entry on oeis.org

3, 4, 8, 10, 20, 22, 32, 34, 40, 56, 56, 70, 80, 82, 88, 100, 116, 116, 130, 140, 140, 154, 160, 170, 190, 200, 202, 212, 214, 212, 250, 256, 272, 268, 296, 296, 308, 322, 328, 340, 356, 352, 380, 382, 392, 386, 410, 442, 452, 454, 460, 476, 472, 496, 508, 520

Offset: 1

Giovanni Teofilatto, Sep 30 2004

Except for the initial term, a(n)=={2, 4} mod 6.

Not monotonic: a(29) = 214 > 212 = a(30), a(33) = 272 > 268 = a(34), etc. - Charles R Greathouse IV, Jun 03 2013

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A001223, A001748, A047235, A100021.

ListConvolve[{-1,3},Prime[Range[100]]] (* Paolo Xausa, Nov 02 2023 *)

a(n) = 3*prime(n) - prime(n+1) \\ Michel Marcus, Jun 03 2013

a(n) = A001043(n) - 2*A001223(n).
a(n) = 3*A000040(n)-A000040(n+1) = A001748(n)-A000040(n+1) = A001747(n+1)-A001223(n). - R. J. Mathar, Apr 22 2010
a(n) ~ 2n log n. - Charles R Greathouse IV, Jun 03 2013
a(n) = A100021(n) + 3. - Hugo Pfoertner, Nov 02 2023
a(n) = A062234(n) + A000040(n). - Anthony S. Wright, Feb 19 2024

Corrected (116 duplicated) by R. J. Mathar, Apr 22 2010

A102724 Sum of the first n pairs of consecutive primes (for example, a(3) = (2+3) + (3+5) + (5+7) = 25).

Original entry on oeis.org

5, 13, 25, 43, 67, 97, 133, 175, 227, 287, 355, 433, 517, 607, 707, 819, 939, 1067, 1205, 1349, 1501, 1663, 1835, 2021, 2219, 2423, 2633, 2849, 3071, 3311, 3569, 3837, 4113, 4401, 4701, 5009, 5329, 5659, 5999, 6351, 6711, 7083, 7467, 7857, 8253, 8663, 9097

Offset: 1

Giovanni Teofilatto, Feb 07 2005

Partial sums of A001043.

			a(1) = 5 = (2+3).
a(2) = 13 = (2+3) + (3+5).
a(3) = 25 = (2+3) + (3+5) + (5+7).

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

Cf. A001043, A102725, A102729, A378569.

Table[Sum[Prime[i] + Prime[i + 1], {i, n}], {n, 47}] (* Ray Chandler, Feb 12 2005 *)
Table[2Sum[Prime[i], {i, n}] - (2 + Prime[n]), {n, 2, 50}] (* Alonso del Arte, Apr 26 2016 *)
Accumulate[Total/@Partition[Prime[Range[50]],2,1]] (* Harvey P. Dale, Apr 13 2019 *)

a(n) = 2*Sum_{i=1..n} prime(i) - (2 + prime(n)). - Alonso del Arte, Apr 21 2016

a(n) = 3*n*(n-1)+7 for n = 2, ..., 8; cf. A378569. - M. F. Hasler, Feb 04 2025

Edited and extended by Ray Chandler, Feb 12 2005

Better definition from Alonso del Arte, Apr 26 2016

A105936 Numbers that are the product of exactly 3 primes and are of the form prime(n) + prime(n+1).

Original entry on oeis.org

8, 12, 18, 30, 42, 52, 68, 78, 138, 172, 186, 222, 258, 268, 410, 434, 508, 548, 618, 668, 762, 772, 786, 892, 906, 946, 978, 1002, 1030, 1132, 1334, 1374, 1446, 1542, 1606, 1758, 1866, 1878, 1948, 2006, 2022, 2252, 2334, 2414, 2452, 2468, 2486, 2572, 2588

Offset: 1

Giovanni Teofilatto, Apr 27 2005

Minimal triprimes (A014612) of the form prime(n) + prime(n+1).
Intersection of A001043 and A014612:
a(1) = A001043(2) = A014612(1),
a(2) = A001043(2) = A014612(2). - Zak Seidov, Jan 31 2017

			a(2) = 12 because 12 = 5 + 7 = 3*2^2;
a(3) = 18 because 18 = 7 + 11 = 2*3^2.

Zak Seidov, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A001043, A014612.

Select[Range[8,10000,2], 3 == PrimeOmega[#] && NextPrime[#/2] + NextPrime[#/2, -1] == # &] (* Zak Seidov, Jan 31 2017 *)

list(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2,min(p,lim\p\2), my(t=2*p*q); if(precprime(p*q)+nextprime(p*q)==t, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

a(n) is equal to the product of 3 primes if the arithmetic mean of prime(n) and prime(n+1) is a semiprime.

More terms from Reinhard Zumkeller, May 03 2005

cp's OEIS Frontend

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

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A096278 Sums of successive sums of successive sums of successive primes.

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A098037 Number of prime divisors, counted with multiplicity, of the sum of two consecutive primes.

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A226524 Cubes which are the sum of two consecutive primes.

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A298463 The first of two consecutive pentagonal numbers the sum of which is equal to the sum of two consecutive primes.

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A092163 a(n) = prime(2n) + prime(2n+1).

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A096279 Sums of successive sums of successive sums of successive sums of successive primes.

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