A175309 a(n) = the smallest prime prime(k) such that prime(k+j) - prime(k+j-1) = prime(n+k+1-j) - prime(n+k-j) for all j with 1 <= j <= n.

2, 3, 5, 18713, 5, 683747, 17, 98303867, 13, 60335249851, 137, 1169769749111, 8021749, 3945769040698829, 1071065111, 159067808851610411, 1613902553, 6919940122097246303, 1797595814863

Offset: 1

Author

Leroy Quet, Mar 27 2010

Keywords

• nonn
• more

Comments

From M. F. Hasler, Apr 02 2010: (Start)

Also: Start of the first sequence of n+1 consecutive primes symmetrically distributed w.r.t. their barycenter, e.g., [2,3], [3,5,7], [5,7,11,13], [18713, 18719, 18731, 18743, 18749]. With this definition, it would make sense to prefix the sequence with an initial term a(0)=2.

Sequence A081235 (or A055382, which is essentially the same) consists of every other term of this sequence. (End)

a(19) = 1797595814863, a(21) = 633925574060671, a(23) = 22930603692243271. - Tomáš Brada, May 25 2020

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..19
• BOINC project to search all up to 2^64
• Symmetric Prime Tuples, SPT test project

Crossrefs

Cf. A006562, A051795, A081235, A081415, A096710, A055382.

Programs

Mathematica

A175309[n_] := Module[{k},
   k = 1; While[! AllTrue[Range[n], Prime[k+#] - Prime[k+#-1] ==
        Prime[n+k+1-#] - Prime[n+k-#] &], k++]; Return[Prime[k]]];
Table[A175309[n], {n, 1, 7}]  (* Robert Price, Mar 27 2019 *)

PARI

a(n)={ my( last=vector(n++,i,prime(i)), m, i=Mod(n-2,n)); forprime(p=last[n],default(primelimit), m=last[1+lift(i+2)]+last[1+lift(i++)]=p; for( j=1,n\2, last[1+lift(i-j)]+last[1+lift(i+j+1)]==m || next(2)); return( last[1+lift(i+1)])) } \\ M. F. Hasler, Apr 02 2010

PARI

isok(p, n) = {my(k=primepi(p)); for (j=1, n, if (prime(k+j) - prime(k+j-1) != prime(n+k+1-j) - prime(n+k-j), return (0));); return (1);} \\ Michel Marcus, Apr 08 2017

Formula

a(2n-1) = A081235(n) (= A055382(n) for n>1). - M. F. Hasler, Apr 02 2010

Extensions

Terms through a(12) were calculated by (in alphabetical order) Franklin T. Adams-Watters, Hans Havermann and D. S. McNeil

Minor edits by N. J. A. Sloane, Apr 02 2010

a(14) from Dmitry Petukhov, added by Max Alekseyev, Nov 03 2014

a(16) from BOINC project, added by Dmitry Petukhov, Apr 06 2017

a(18)-a(19) from SPT test project, added by Dmitry Petukhov, Mar 16 2025