A070934 -id:A070934 - cp's OEIS frontend

A070281 Smallest prime which is the sum of n consecutive primes, or 0 if no such prime exists.

2, 5, 23, 17, 53, 41, 197, 0, 127, 0, 233, 197, 691, 281, 379, 0, 499, 0, 857, 0, 953, 0, 1151, 0, 1259, 0, 1583, 0, 2099, 0, 2399, 0, 2417, 0, 2579, 0, 2909, 0, 3803, 0, 3821, 0, 4217, 0, 4651, 0, 5107, 0, 5813, 0, 6829, 0, 6079, 0, 6599, 0, 14153, 0, 10091, 7699

Offset: 1

Amarnath Murthy, May 07 2002

nonn

Sequence A013918 lists the nonzero numbers occurring at even n.

			a(60) = 7699 because Sum_{k=1..60} prime(k) = 7699 and 7699 is the smallest possible prime formed by summing 60 consecutive primes. - _Sean A. Irvine_, Jun 07 2024

Cf. A070934.

f[n_] := Block[{k = 1, s},If[Mod[n, 2] == 0,s = Sum[Prime[i], {i, k, k + n - 1}];If[PrimeQ[s], s, 0],While[s = Sum[Prime[i], {i, k, k + n - 1}] ; ! PrimeQ[s], k++ ];s]];Table[f[n], {n, 65}] (* Ray Chandler, Sep 27 2006 *)

Corrected and extended by Jim Nastos, Jun 15 2002
Extended by Ray Chandler, Sep 27 2006
a(60) corrected by Giovanni Resta, May 31 2017
Original a(60) restored by Sean A. Irvine, Jun 07 2024

A356477 a(n) is the start of the first sequence of 2n+1 consecutive primes p_1, p_2, ..., p_(2n+1) such that p_1p_2 + p_2p_3 + ... + p_(2n)p_(2n+1) + p_(2n+1)*p_1 is prime.

Original entry on oeis.org

2, 19, 19, 2, 23, 2, 7, 7, 2, 5, 113, 5, 29, 13, 67, 53, 11, 11, 5, 23, 7, 43, 5, 2, 31, 73, 13, 3, 89, 5, 11, 3, 89, 31, 43, 2, 37, 2, 23, 7, 11, 19, 43, 23, 5, 2, 23, 3, 29, 5, 17, 3, 31, 29, 53, 29, 7, 13, 73, 3, 5, 43, 29, 17, 5, 37, 19, 11, 71, 7, 2, 43, 13, 19, 2, 59, 7, 29, 113, 13, 5, 11

Offset: 1

J. M. Bergot and Robert Israel, Aug 08 2022

nonn

			a(2) = 19 because 19 is the start of the 2*2+1 = 5 consecutive primes 19, 23, 29, 31, 37 with 19*23 + 23*29 + 29*31 + 31*37 + 37*19 = 3853 prime, and no earlier 5-tuple of consecutive primes works.

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

Cf. A070934, A356471, A356475.

f:= proc(m) local P,x,i,n;
  n:= 2*m+1;
  P:= Vector(n,ithprime);
do
   x:= add(P[i]*P[i+1],i=1..n-1)+P[n]*P[1];
   if isprime(x) then return P[1] fi;
   P[1..n-1]:= P[2..n];
   P[n]:= nextprime(P[n]);
od
end proc:
map(f, [$1..100]);

from sympy import isprime, nextprime, prime, primerange
def a(n):
    p = list(primerange(1, prime(2*n+1)+1))
    while True:
        if isprime(sum(p[i]*p[i+1] for i in range(len(p)-1))+p[-1]*p[0]):
            return p[0]
        p = p[1:] + [nextprime(p[-1])]
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Aug 08 2022

A127492 Indices m of primes such that Sum_{k=0..2, k

Original entry on oeis.org

2, 10, 17, 49, 71, 72, 75, 145, 161, 167, 170, 184, 244, 250, 257, 266, 267, 282, 286, 301, 307, 325, 343, 391, 405, 429, 450, 537, 556, 561, 584, 685, 710, 743, 790, 835, 861, 904, 928, 953

Offset: 1

Artur Jasinski, Jan 16 2007

Let p_0 .. p_4 be five consecutive primes, starting with the m-th prime. The index m is in the sequence if the absolute value [x^0] of the polynomial (x-p_0)*[(x-p_1)*(x-p_2) + (x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_1)*[(x-p_2)*(x-p_3) + (x-p_3)*(x-p_4)] + (x-p_2)*(x-p_3)*(x-p_4) is two times a prime. The correspondence with A127491: the coefficient [x^2] of the polynomial (x-p_0)*(x-p_1)*..*(x-p_4) is the sum of 10 products of a set of 3 out of the 5 primes. Here the sum is restricted to the 6 products where the two largest of the 3 primes are consecutive. - R. J. Mathar, Apr 23 2023

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338, A127339, A127340, A127341, A127342, A127343, A127345, A127346, A127347, A127348, A127349, A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127489, A127490, A127491.

isA127492 := proc(k)
    local x,j ;
    (x-ithprime(k))* mul( x-ithprime(k+j),j=1..2)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=2..3)
    +(x-ithprime(k+1))* mul( x-ithprime(k+j),j=3..4)
    +(x-ithprime(k+2))* mul( x-ithprime(k+j),j=3..4) ;
    p := abs(coeff(expand(%/2),x,0)) ;
    if type(p,'integer') then
        isprime(p) ;
    else
        false ;
    end if ;
end proc:
for k from 1 to 900 do
    if isA127492(k) then
        printf("%a,",k) ;
    end if ;
end do: # R. J. Mathar, Apr 23 2023

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2] + Prime[x] Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3] + Prime[x + 1] Prime[x + 3]Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])/2], AppendTo[a, x]], {x, 1, 1000}]; a
prQ[{a_,b_,c_,d_,e_}]:=PrimeQ[(a b c+a c d+a d e+b c d+b d e+c d e)/2]; PrimePi/@Select[ Partition[ Prime[Range[1000]],5,1],prQ][[;;,1]] (* Harvey P. Dale, Apr 21 2023 *)

Definition simplified by R. J. Mathar, Apr 23 2023

Edited by Jon E. Schoenfield, Jul 23 2023

A216004 Primes that are the sum of 2001 consecutive primes.

Original entry on oeis.org

16344247, 16588633, 16711217, 17416457, 17700139, 17806721, 17860039, 17895613, 18091313, 18144727, 18483739, 18573209, 18698791, 19040773, 19384609, 19529849, 19620719, 19748129, 19784543, 19802759, 20421971, 20476777, 20531593, 20806141, 21283169, 21356563

Offset: 1

Syed Iddi Hasan, Aug 30 2012

nonn

Corresponding initial terms of 2001-tuples: 7, 61, 97, 313, 419, 449, 463, 479, 563, 577, 691, 733, 773, 911, 1039, 1093, 1123, 1187, 1201, 1213, 1459, 1483, 1493, 1607. The indices of these primes: 4, 18, 25, 65, 81, 87, 90, 92, 103, 106, 125, 130, 137, 156, 175, 183, 188, 195, 197, 198, 232, 235, 238, 253. - Zak Seidov, Feb 27 2013

			a(1) = sum(prime(k), k = 4..2004),  a(2) = sum(prime(k), k = 18..2018),... - _Zak Seidov_, Feb 27 2013

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

Cf. A070934, A082244, A215991.

m = 2001; a = 2; b = Prime[m]; s = Sum[Prime[k], {k, m}]; Reap[Do[If[PrimeQ[s], Sow[s]]; b = NextPrime[b]; s = s - a + b; a = NextPrime[a], {m}]][[2, 1]] (* Zak Seidov, Feb 27 2013 *)
Select[Total/@Partition[Prime[Range[2500]],2001,1],PrimeQ] (* Harvey P. Dale, Mar 13 2018 *)

A122654 Smallest prime p such that p^2 equal to the sum of 2n+1 consecutive odd primes, or 1 if such a prime does not exist.

Original entry on oeis.org

7, 31, 13, 29, 107, 293, 821, 379, 293, 83, 31, 241, 37, 653, 43, 211, 73, 1123, 311, 1567, 1607, 929, 233, 4801, 601, 5087, 307, 163, 709, 1021, 983, 191, 1327, 241, 5443, 157, 277, 151, 743, 4651, 32371, 1493, 1373, 8521, 683, 7919, 947, 1279, 10847, 1613

Offset: 1

Alexander Adamchuk, Sep 21 2006

nonn

			a(1) = 7 because A122560[1] = 7 and 7^2 = 49 = 13 + 17 + 19 = p(6) + p(7) + p(8).

Cf. A122560, A070934.

More terms from Don Reble, Sep 22 2006

A127493 Indices k such that the coefficient [x^1] of the polynomial Product_{j=0..4} (x-prime(k+j)) is prime.

Original entry on oeis.org

1, 5, 8, 9, 22, 29, 45, 49, 60, 69, 87, 89, 90, 107, 114, 124, 125, 131, 134, 138, 145, 156, 171, 183, 188, 191, 203, 204, 207, 212, 219, 255, 261, 290, 298, 303, 329, 330, 343, 344, 349, 354, 378, 397, 398, 400, 403, 454, 456, 466, 474, 515, 549, 560, 570, 578

Offset: 1

Artur Jasinski, Jan 16 2007

nonn

A fifth-order polynomial with 5 roots which are the five consecutive primes from prime(k) onward is defined by Product_{j=0..4} (x-prime(k+j)). The sequence is a catalog of the cases where the coefficient of its linear term is prime.

Indices k such that e4(prime(k), prime(k+1), ..., prime(k+4)) is prime, where e4 is the elementary symmetric polynomial summing all products of four variables. - Charles R Greathouse IV, Jun 15 2015

			For k=2, the polynomial is (x-3)*(x-5)*(x-7)*(x-11)*(x-13) = x^5-39*x^4+574*x^3-3954*x^2+12673*x-15015, where 12673 is not prime, so k=2 is not in the sequence.
For k=5, the polynomial is x^5-83*x^4+2710*x^3-43490*x^2+342889*x-1062347, where 342889 is prime, so k=5 is in the sequence.

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

Cf. A001043, A034961, A034963, A034964, A127333-A127343, A127345-A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301-A046303, A046324-A046327, A127489, A127491, A127492, A024449.

isA127493 := proc(k)
    local x,j ;
    mul( x-ithprime(k+j),j=0..4) ;
    expand(%) ;
    isprime(coeff(%,x,1)) ;
end proc:
A127493 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA127493(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A127493(n),n=1..60) ; # R. J. Mathar, Apr 23 2023

a = {}; Do[If[PrimeQ[(Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 3] + Prime[x] Prime[x + 2]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 3]Prime[x + 4] + Prime[x] Prime[x + 1]Prime[x + 2]Prime[x + 4] + Prime[x + 1] Prime[x + 2]Prime[x + 3]Prime[x + 4])], AppendTo[a, x]], {x, 1, 1000}]; a

e4(v)=sum(i=1,#v-3, v[i]*sum(j=i+1,#v-2, v[j]*sum(k=j+1,#v-1, v[k]*vecsum(v[k+1..#v]))))
pr(p, n)=my(v=vector(n)); v[1]=p; for(i=2,#v, v[i]=nextprime(v[i-1]+1)); v
is(n,p=prime(n))=isprime(e4(pr(p,5)))
v=List(); n=0; forprime(p=2,1e4, if(is(n++,p), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 15 2015

Definition and comment rephrased and examples added by R. J. Mathar, Oct 01 2009

A125270 Coefficient of x^2 in polynomial whose zeros are 5 consecutive primes starting with the n-th prime.

Original entry on oeis.org

1358, 3954, 10478, 22210, 43490, 78014, 129530, 206650, 324350, 466270, 621466, 853742, 1132130, 1436690, 1870850, 2388050, 2886370, 3440410, 4133410, 4904906, 5926654, 7195670, 8425430, 9792950, 11040910, 12098990, 13917898, 16097810

Offset: 1

Artur Jasinski, Jan 16 2007

nonn

Sums of all distinct products of 3 out of 5 consecutive primes, starting with the n-th prime; value of 3rd elementary symmetric function on the 5 consecutive primes.

Cf. A001043, A034961, A034963, A034964, A127333, A127334, A127335, A127336, A127337, A127338, A127339, A127340, A127341, A127342, A127343, A127345, A127346, A127347, A127348, A127349, A127351, A037171, A034962, A034965, A082246, A082251, A070934, A006094, A046301, A046302, A046303, A046324, A046325, A046326, A046327, A127489.

a = {}; Do[AppendTo[a, (Prime[x] Prime[x + 1] Prime[x + 2] + Prime[x] Prime[x + 1] Prime[x + 3] + Prime[x] Prime[x + 1] Prime[x + 4] + Prime[x] Prime[x + 2] Prime[x + 3] + Prime[x] Prime[x + 2] Prime[x + 4] + Prime[x] Prime[x + 3] Prime[x + 4] + Prime[x + 1] Prime[x + 2] Prime[x + 3] + Prime[x + 1] Prime[x + 2] Prime[x + 4] + Prime[x + 1] Prime[x + 3] Prime[x + 4] + Prime[x + 2] Prime[x + 3] Prime[x + 4])], {x, 1, 100}]; a
fcp[{p_,q_,r_,s_,t_}]:=p*q(r+s+t)+(p+q)r(s+t)+(p+q+r)s*t; fcp/@Partition[ Prime[ Range[40]],5,1] (* Harvey P. Dale, Sep 05 2014 *)

Let p = Prime(n), q = Prime(n+1), r = Prime(n+2), s = Prime(n+3) and t = Prime(n+4). Then a(n) = p q (r+s+t) + (p + q) r (s + t) + (p + q + r) s t.

Edited and corrected by Franklin T. Adams-Watters, Jan 23 2007

cp's OEIS Frontend

A070281 Smallest prime which is the sum of n consecutive primes, or 0 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A356477 a(n) is the start of the first sequence of 2*n+1 consecutive primes p_1, p_2, ..., p_(2*n+1) such that p_1*p_2 + p_2*p_3 + ... + p_(2*n)*p_(2*n+1) + p_(2*n+1)*p_1 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Python

A127492 Indices m of primes such that Sum_{k=0..2, k

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A216004 Primes that are the sum of 2001 consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A122654 Smallest prime p such that p^2 equal to the sum of 2n+1 consecutive odd primes, or 1 if such a prime does not exist.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A127493 Indices k such that the coefficient [x^1] of the polynomial Product_{j=0..4} (x-prime(k+j)) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A125270 Coefficient of x^2 in polynomial whose zeros are 5 consecutive primes starting with the n-th prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A356477 a(n) is the start of the first sequence of 2n+1 consecutive primes p_1, p_2, ..., p_(2n+1) such that p_1p_2 + p_2p_3 + ... + p_(2n)p_(2n+1) + p_(2n+1)*p_1 is prime.