A214196 Unique terms in sequence A210144.

2, 3, 5, 11, 23, 29, 37, 41, 47, 73, 131, 151, 199, 223, 271, 281, 353, 457, 641, 643, 659, 1259, 1531, 1747, 1951, 2671, 2953, 4259, 4967, 5419, 5839, 7013, 7963, 11261, 12653, 15733, 16189, 18367, 19237, 29129, 32381, 33161, 33247, 57653, 61723, 63823, 66739

Offset: 1

Author

N. J. A. Sloane, Jul 07 2012

Keywords

• nonn

Comments

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

In Sun (2012), these numbers are called "Primes of the first kind".

Conjecture: all the terms are prime. See Conjecture 1.5(i) in Sun 2012. - Jason Yuen, Feb 25 2024

Links

• Jason Yuen, Table of n, a(n) for n = 1..128
• Zhi-Wei Sun, On functions taking only prime values, arXiv preprint arXiv:1202.6589 [math.NT], 2012; see p. 5-6.

Crossrefs

Cf. A210144, A214197.

Programs

Maple

A214196 := proc(n)
        local m ,i,j,ddvs;
        for m from 2 do
                ddvs := false ;
                for i from 1 to n-1 do
                        for j from i+1 to n do
                                if (A002110(j)-A002110(i)) mod m = 0 then
                                        ddvs := true;
                                        break;
                                end if;
                        end do:
                        if ddvs then
                                break;
                        end if;
                end do:
                if ddvs = false then
                        return m;
                end if;
        end do:
end proc:
# loop generates m multiples times (pipe through 'uniq')
for n from 1 do
        printf("%d,\n",A214196(n)) ;
end do: # R. J. Mathar, Jul 08 2012

Mathematica

primorial[n_] := primorial[n] = Product[Prime[i], {i, 1, n}];
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]]]];
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 *)

Extensions

a(28)-a(34) from Jean-François Alcover, Jan 20 2018

Definition simplified and more terms from Jason Yuen, Feb 24 2024