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%I #28 Mar 23 2024 21:09:20
%S 2,3,5,11,23,29,37,41,47,73,131,151,199,223,271,281,353,457,641,643,
%T 659,1259,1531,1747,1951,2671,2953,4259,4967,5419,5839,7013,7963,
%U 11261,12653,15733,16189,18367,19237,29129,32381,33161,33247,57653,61723,63823,66739
%N Unique terms in sequence A210144.
%C The sequence is the set of numbers m which are the minimum m for some triple 1 <= i < j <= k such that m divides none of the differences A002110(i)-A002110(j). - _R. J. Mathar_, Jul 08 2012
%C In Sun (2012), these numbers are called "Primes of the first kind".
%C Conjecture: all the terms are prime. See Conjecture 1.5(i) in Sun 2012. - _Jason Yuen_, Feb 25 2024
%H Jason Yuen, <a href="/A214196/b214196.txt">Table of n, a(n) for n = 1..128</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1202.6589">On functions taking only prime values</a>, arXiv preprint arXiv:1202.6589 [math.NT], 2012; see p. 5-6.
%p A214196 := proc(n)
%p local m ,i,j,ddvs;
%p for m from 2 do
%p ddvs := false ;
%p for i from 1 to n-1 do
%p for j from i+1 to n do
%p if (A002110(j)-A002110(i)) mod m = 0 then
%p ddvs := true;
%p break;
%p end if;
%p end do:
%p if ddvs then
%p break;
%p end if;
%p end do:
%p if ddvs = false then
%p return m;
%p end if;
%p end do:
%p end proc:
%p # loop generates m multiples times (pipe through 'uniq')
%p for n from 1 do
%p printf("%d,\n",A214196(n)) ;
%p end do: # _R. J. Mathar_, Jul 08 2012
%t primorial[n_] := primorial[n] = Product[Prime[i], {i, 1, n}];
%t p[0] = 1; p[n_] := p[n] = Module[{m, i, j, ddvs}, For[m = 2, True, m++, ddvs = False ; For[i = 1, i <= n - 1, i++, For[j = i + 1, j <= n, j++, If[Mod[primorial[j] - primorial[i], m] == 0, ddvs = True; Break[]]]; If[ddvs, Break[]]]; If[ddvs == False, Return[m]]]];
%t A214196 = Reap[n = k = 1; While[n <= 400, If[p[n] != p[n - 1], a[k] = p[n]; Print[n, " a(", k, ") = ", a[k]]; Sow[a[k]]; k++]; n++]][[2, 1]] (* _Jean-François Alcover_, Jan 20 2018, after _R. J. Mathar_ *)
%Y Cf. A210144, A214197.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Jul 07 2012
%E a(28)-a(34) from _Jean-François Alcover_, Jan 20 2018
%E Definition simplified and more terms from _Jason Yuen_, Feb 24 2024
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