A001043 -id:A001043 - cp's OEIS frontend

A127339 Numbers that are the sum of 12 consecutive primes.

197, 236, 276, 318, 364, 412, 460, 510, 562, 612, 662, 714, 766, 822, 880, 936, 990, 1040, 1092, 1152, 1212, 1276, 1336, 1402, 1464, 1524, 1586, 1650, 1716, 1786, 1854, 1918, 1980, 2040, 2100, 2162, 2234, 2304, 2370, 2436, 2502, 2564, 2634, 2700, 2770

Offset: 1

Artur Jasinski, Jan 11 2007

nonn

a(n) = absolute value of coefficient of x^11 of the polynomial Product_{j=0..11} (x - prime(n+j)) of degree 12; the roots of this polynomial are prime(n), ..., prime(n+11).

Cf. A011974, A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338.

[&+[ NthPrime(n+k): k in [0..11] ]: n in [1..100] ]; // Vincenzo Librandi, Apr 03 2011

a = {}; Do[AppendTo[a, Sum[Prime[x + n], {n, 0, 11}]], {x, 1, 50}]; a
Total/@Partition[Prime[Range[60]],12,1] (* Harvey P. Dale, May 05 2018 *)

{m=45;k=12;for(n=1,m,print1(a=sum(j=0,k-1,prime(n+j)),","))} \\ Klaus Brockhaus, Jan 13 2007

{m=45;k=12;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

A071215 Number of distinct prime factors of sum of 2 successive primes.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 2, 3, 1, 3, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 4, 3, 2, 3, 3, 2, 4, 3, 3, 3, 3, 3, 4, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 2, 3, 4, 3, 3, 4, 3, 2, 2, 3, 2, 3, 3, 4, 4, 3, 4, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 2, 4, 3

Offset: 1

Labos Elemer, May 17 2002

			Prime(6) = 13 and prime(7) = 17. 13 + 17 = 30 = 2 * 3 * 5, which has three distinct prime factors, hence a(6) = 3.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001043, A001221, A071216, A251609 (greedy inverse).

Table[PrimeNu[Prime[n] + Prime[n + 1]], {n, 105}] (* Jean-François Alcover, Oct 21 2013 *)

A071215(n)=omega(prime(n)+prime(n+1)) \\ M. F. Hasler, Jul 23 2007

a(n) = omega(prime(n) + prime(n + 1)) = A001221(A001043(n)), where omega is the number of distinct prime factors function.

A092738 Primes of the form prime(x)+prime(x+1)+1.

Original entry on oeis.org

13, 19, 31, 37, 43, 53, 61, 79, 101, 113, 139, 163, 173, 199, 211, 223, 241, 269, 277, 331, 353, 373, 397, 457, 463, 509, 521, 541, 577, 601, 619, 631, 727, 773, 787, 811, 829, 853, 883, 907, 919, 947, 967, 991, 1013, 1031, 1181, 1193, 1201, 1231, 1237

Offset: 1

Jorge Coveiro, Apr 12 2004

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

Cf. A001043, A072669.

f[n_] := Prime[n] + Prime[n + 1] + 1; f[ # ] & /@ Select[ Range[120], PrimeQ[ f[ # ]] &] (* Robert G. Wilson v, Apr 14 2004 *)

list(lim)=my(v=List(),p=2,t);forprime(q=3,,t=p+q+1;if(t>lim,return(Vec(v)));if(isprime(t),listput(v,t))) \\ Charles R Greathouse IV, Mar 19 2013

More terms from Robert G. Wilson v, Apr 14 2004

A111163 Triangular numbers that are sums of two consecutive primes.

Original entry on oeis.org

36, 78, 120, 210, 276, 300, 630, 946, 990, 1770, 1830, 2556, 2850, 3240, 3570, 4278, 4950, 5460, 8256, 9870, 10878, 11026, 12090, 12720, 20100, 20910, 23436, 26796, 31626, 34980, 41616, 43660, 46056, 55278, 56616, 57630, 59340, 66066, 73920

Offset: 1

Joseph L. Pe, Oct 20 2005

nonn

Intersection of A000217 and A001043. - Michel Marcus, May 18 2015

			36 = 8(8+1)/2 = 17 + 19. Therefore 36 belongs to the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..472 from K. D. Bajpai, n = 473..1000 from T. D. Noe)

Cf. A000217 (triangular numbers), A001043 (sums of consecutive primes).

P:=proc()local i,a,b,c;c:=1;for i from 1 to 1200000 do;
a:=ithprime(i)+ithprime(i+1);b:=(-1+(sqrt(8*a+1)))/2;
if  b=floor(b) then lprint(c,a);c:=c+1;fi; od;end:P();
# K. D. Bajpai, Jun 30 2013

Select[Table[Prime[n] + Prime[n + 1], {n, 4500}], IntegerQ[Sqrt[1 + 8# ]] &] (* Ray Chandler, Oct 22 2005 *)
t = {}; n = 0; While[Length[t] < 100, n++; s = Prime[n] + Prime[n + 1]; If[TriangularQ[s], AppendTo[t, s]]]; t (* T. D. Noe, Jun 30 2013 *)

{p=2; ct=0; while(ct<66, q=nextprime(p+1); s=p+q; if( issquare(8*s+1), print1(s, ", "); ct++); p=q)} \\ Klaus Brockhaus, Oct 22 2005

from sympy import nextprime as np
from sympy import prevprime as pp
n=1
t=n*(n+1)//2
while t>0:
    if t%2==0 and t>3:
        i=t//2
        p=pp(i); q=np(p)
        if p+q==t :
            print(t)
    n=n+1
    t=n*(n+1)//2 # Abhiram R Devesh, May 17 2015

from sympy import prevprime,nextprime,isprime
A111163_list = [n*(n+1)//2 for n in range(3,10**4) if not isprime(n*(n+1)//4) and prevprime(n*(n+1)//4)+nextprime(n*(n+1)//4) == n*(n+1)//2] # Chai Wah Wu, Feb 11 2018

Extended by Ray Chandler and Klaus Brockhaus, Oct 22 2005

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Original entry on oeis.org

6, 8, 10, 10, 12, 14, 14, 16, 18, 22, 16, 18, 20, 24, 26, 20, 22, 24, 28, 30, 34, 22, 24, 26, 30, 32, 36, 38, 26, 28, 30, 34, 36, 40, 42, 46, 32, 34, 36, 40, 42, 46, 48, 52, 58, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74, 44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82

Offset: 1

Reinhard Zumkeller, Feb 06 2006

T(n,k) = 2*A065305(n,k) = A065342(n+1,k+1);
Row sums give A116367; central terms give A116368;
T(n,1) = A113935(n+1);
T(n,n-2) = A048448(n) for n>2;
T(n,n-1) = A001043(n) for n>1;
T(n,n) = A001747(n+2) = A100484(n+1).

			Triangle begins:
  6;
  8,  10;
  10, 12, 14;
  14, 16, 18, 22;
  16, 18, 20, 24, 26;
  20, 22, 24, 28, 30, 34;
  22, 24, 26, 30, 32, 36, 38;
  26, 28, 30, 34, 36, 40, 42, 46;
  32, 34, 36, 40, 42, 46, 48, 52, 58;
  34, 36, 38, 42, 44, 48, 50, 54, 60, 62;
  40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74;
  44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82; etc. - _Bruno Berselli_, Aug 16 2013

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A001043, A001747, A048448, A065305, A065342, A100484, A113935, A116367, A116368.

[NthPrime(n+1)+NthPrime(k+1): k in [1..n], n in [1..15]]; // Bruno Berselli, Aug 16 2013

Table[Prime[n+1] + Prime[k+1], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, May 12 2019 *)

{T(n,k) = prime(n+1) + prime(k+1)}; \\ G. C. Greubel, May 12 2019

[[nth_prime(n+1) + nth_prime(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 12 2019

T(n,k) = prime(n+1) + prime(k+1), 1 <= k <= n.

A163487 Primes p such that 6*p is the sum of two consecutive primes.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 31, 37, 43, 103, 127, 131, 151, 163, 167, 229, 241, 257, 293, 311, 313, 337, 389, 433, 509, 521, 523, 613, 647, 661, 719, 739, 743, 757, 797, 821, 887, 937, 953, 971, 1013, 1033, 1063, 1151, 1153, 1217, 1283, 1303, 1307, 1319, 1373, 1451

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 28 2009

nonn

			2*6=12=5+7, 3*6=18=7+11, 5*6=30=13+17, ..

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

Cf. A001043, A118134.

Primes:= select(isprime, [seq(i,i=3..10^4, 2)]):
select(t -> t::integer and isprime(t), (Primes[1..-2]+Primes[2..-1])/6); # Robert Israel, Jun 19 2018

Select[ListConvolve[{1,1},Prime[Range[1000]]]/6,PrimeQ] (* Paolo Xausa, Nov 03 2023 *)

Edited by N. J. A. Sloane, Aug 08 2009

A036441 a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2.

Original entry on oeis.org

2, 4, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 70, 77, 88, 99, 110, 121, 132, 143, 156, 169, 182, 195, 208, 221, 238, 255, 272, 289, 306, 323, 342, 361, 380, 399, 418, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 696, 725, 754, 783, 812, 841, 870

Offset: 1

Frederick Magata (frederick.magata(AT)uni-muenster.de)

a(n) satisfies the following inequality: (1/4)*(n^2 + 3*n + 1) <= a(n) <= (1/4)*(n+2)^2. [Corrected by M. F. Hasler, Apr 08 2015]
The present sequence is the special case a(n) = a(2,n) with a more general a(m, n) := a(m, n-1) + gpf(a(m, n-1)), a(m, 1) := m, where gpf(x) := "greatest prime factor of x" = A006530(x). Also a(a(r,k), n) = a(r,n+k-1), for all n,k in N\{0} and all r in N\{0,1}; a(prime(k), n) = a(prime(i), n + prime(k) - prime(i)), for all k,i,n in N\{0}, with k >= i, n >= prime(k-1) and with prime(x) := x-th prime.
Essentially the same as A076271 and A180107, cf. formula.

			a(2,2) = 4 because 2 + gpf(2) = 2 + 2 = 4;
a(2,3) = 6 because 4 + gpf(4) = 4 + 2 = 6.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A006530. See A076271 and A180107 for other versions.
Cf. A001043, A070229.
Cf. A123581.
Partial sums of A076973.

a036441 n = a036441_list !! (n-1)
a036441_list = tail a076271_list
-- Reinhard Zumkeller, Nov 08 2015, Nov 14 2011

f[n_]:=Last[First/@FactorInteger[n]];Join[{a=2},Table[a+=f[a],{n,2,100}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
NestList[#+FactorInteger[#][[-1,1]]&,2,60] (* Harvey P. Dale, Dec 02 2012 *)

a(n)=(n+2-if(n\2+1<(p=nextprime(n\2+1))&&n+1M. F. Hasler, Apr 08 2015

a(n) = p(m)*(n+2-p(m)), where p(k) is the k-th prime and m is the smallest index such that n+2 <= p(m) + p(m+1). - Max Alekseyev, Oct 21 2008
a(n) = A076271(n+1) = A180107(n+2). - M. F. Hasler, Apr 08 2015
a(n+1) = A070229(a(n)). - Reinhard Zumkeller, Nov 07 2015

Better description from Reinhard Zumkeller, Feb 04 2002

Edited by M. F. Hasler, Apr 08 2015

A061802 Sum of n-th row of triangle of primes: 2; 2 3 2; 2 3 5 3 2; 2 3 5 7 5 3 2; ...; where n-th row contains 2n+1 terms.

Original entry on oeis.org

2, 7, 15, 27, 45, 69, 99, 135, 177, 229, 289, 357, 435, 519, 609, 709, 821, 941, 1069, 1207, 1351, 1503, 1665, 1837, 2023, 2221, 2425, 2635, 2851, 3073, 3313, 3571, 3839, 4115, 4403, 4703, 5011, 5331, 5661, 6001, 6353, 6713, 7085, 7469, 7859, 8255, 8665

Offset: 0

Amarnath Murthy, May 28 2001

Row sums of A138143. - Omar E. Pol, Feb 13 2014

For n = 3..9, a(n) = 3*(n^2 - 3*n + 5). - Nicholas Drozd, Apr 10 2021

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Harry J. Smith)

Cf. A001043 (first differences), A007504, A138143.

Partial sums of A011974.

Accumulate[Join[{2},ListConvolve[{1,1},Prime[Range[100]]]]] (* Paolo Xausa, Oct 31 2023 *)

{ n=-1; a=q=0; forprime (p=2, prime(1001), write("b061802.txt", n++, " ", a+=p + q); q=p ) } \\ Harry J. Smith, Jul 28 2009

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

a(n) = A007504(n) + A007504(n+1) so we have the asymptotic expansion a(n) ~ n^2*log(n). - Henry Bottomley, May 30 2001

More terms from Larry Reeves (larryr(AT)acm.org), May 29 2001

A071220 Numbers n such that prime(n) + prime(n+1) is a cube.

Original entry on oeis.org

2, 28, 1332, 3928, 16886, 157576, 192181, 369440, 378904, 438814, 504718, 539873, 847252, 1291597, 1708511, 1837979, 3416685, 3914319, 5739049, 6021420, 7370101, 7634355, 8608315, 9660008, 10378270, 14797144, 15423070, 18450693

Offset: 1

Labos Elemer, May 17 2002

nonn

The corresponding primes are in A061308; n^3 is a sum of two successive primes in A074925.

Prime(n)+ Prime(n+1) is a square in A064397; n^2 is a sum of two successive primes in A074924;

			28 is in the list because prime(28)+prime(29) = 107+109 =216 = 6^3.
n=1291597: prime(1291597)+prime(1291598) = 344*344*344.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A064397, A074925, A074924, A001043.

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[ If[ n^3 == PrevPrim[Floor[(n^3)/2]] + NextPrim[Floor[(n^3)/2]], Print[ PrimePi[ Floor[(n^3)/2]]]], {n, 2, 10^4}]
Flatten[Position[Total/@Partition[Prime[Range[20000000]],2,1],?(IntegerQ[ Surd[ #,3]]&)]] (* _Harvey P. Dale, May 28 2014 *)

from _future_ import division
from sympy import isprime, prevprime, nextprime, primepi
A071220_list, i = [], 2
while i < 10**6:
    n = i**3
    m = n//2
    if not isprime(m) and prevprime(m) + nextprime(m) == n:
        A071220_list.append(primepi(m))
    i += 1 # Chai Wah Wu, May 31 2017

A001043(x)=m^3 for some m; if p(x+1)+p(x) is a cube, then x is here.

a(n) = primepi(A061308(n)). - Michel Marcus, Oct 24 2014

Edited and extended by Robert G. Wilson v, Oct 07 2002

A076273 a(0) = 1, a(1) = 2; for n>1, a(n) = prime(n)+prime(n-1)-1.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 23, 29, 35, 41, 51, 59, 67, 77, 83, 89, 99, 111, 119, 127, 137, 143, 151, 161, 171, 185, 197, 203, 209, 215, 221, 239, 257, 267, 275, 287, 299, 307, 319, 329, 339, 351, 359, 371, 383, 389, 395, 409, 433, 449, 455, 461, 471, 479, 491, 507, 519, 531

Offset: 0

Reinhard Zumkeller, Oct 04 2002

Least m such that A076272(m) > A076272(m-1) for n>0; a(0)=1.

A076272(a(k)+j) = A008578(k) for k>0 and 0<=j < A075527(k-1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001043.

nxt[{a_,b_}]:={a+1,Prime[a+1]+Prime[a]-1}; Join[{1},Transpose[ NestList[ nxt,{1,2},60]][[2]]] (* or *) Join[{1,2},Total/@Partition[Prime[ Range[ 60]],2,1]-1] (* Harvey P. Dale, Jun 12 2012 *)

a(n)=if(n<1,1,if(n==1,2,prime(n)+prime(n-1)-1)) \\ Lambert Klasen, Jan 14 2005; corrected by Michel Marcus, Nov 05 2023

a(n) = A001043(n-1)-1, n>1. - R. J. Mathar, Jun 04 2020

Simpler description from Vladeta Jovovic, Mar 29 2003

More terms from Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 14 2005

cp's OEIS Frontend

A127339 Numbers that are the sum of 12 consecutive primes.

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A071215 Number of distinct prime factors of sum of 2 successive primes.

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A092738 Primes of the form prime(x)+prime(x+1)+1.

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A111163 Triangular numbers that are sums of two consecutive primes.

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A116366 Triangle read by rows: even numbers as sums of two odd primes.

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A163487 Primes p such that 6*p is the sum of two consecutive primes.

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A036441 a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2.

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