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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.

%I A175309 #77 Aug 08 2025 20:12:33
%S A175309 2,3,5,18713,5,683747,17,98303867,13,60335249851,137,1169769749111,
%T A175309 8021749,3945769040698829,1071065111,159067808851610411,1613902553,
%U A175309 6919940122097246303,1797595814863
%N 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.
%C A175309 From _M. F. Hasler_, Apr 02 2010: (Start)
%C A175309 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.
%C A175309 Sequence A081235 (or A055382, which is essentially the same) consists of every other term of this sequence. (End)
%C A175309 a(19) = 1797595814863, a(21) = 633925574060671, a(23) = 22930603692243271. - _Tomáš Brada_, May 25 2020
%H A175309 Andrew Howroyd, <a href="/A175309/b175309.txt">Table of n, a(n) for n = 1..19</a>
%H A175309 <a href="http://stop.inferia.ru/">BOINC project</a> to search all up to 2^64
%H A175309 Symmetric Prime Tuples, <a href="https://boinc.termit.me/adsl/">SPT test project</a>
%F A175309 a(2n-1) = A081235(n) (= A055382(n) for n>1). - _M. F. Hasler_, Apr 02 2010
%t A175309 A175309[n_] := Module[{k},
%t A175309    k = 1; While[! AllTrue[Range[n], Prime[k+#] - Prime[k+#-1] ==
%t A175309         Prime[n+k+1-#] - Prime[n+k-#] &], k++]; Return[Prime[k]]];
%t A175309 Table[A175309[n], {n, 1, 7}]  (* _Robert Price_, Mar 27 2019 *)
%o A175309 (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
%o A175309 (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
%Y A175309 Cf. A006562, A051795, A081235, A081415, A096710, A055382.
%K A175309 nonn,more
%O A175309 1,1
%A A175309 _Leroy Quet_, Mar 27 2010
%E A175309 Terms through a(12) were calculated by (in alphabetical order) _Franklin T. Adams-Watters_, _Hans Havermann_ and _D. S. McNeil_
%E A175309 Minor edits by _N. J. A. Sloane_, Apr 02 2010
%E A175309 a(14) from _Dmitry Petukhov_, added by _Max Alekseyev_, Nov 03 2014
%E A175309 a(16) from BOINC project, added by _Dmitry Petukhov_, Apr 06 2017
%E A175309 a(18)-a(19) from SPT test project, added by _Dmitry Petukhov_, Mar 16 2025