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A080743 Array read by rows in which n-th row lists orders of elements of Symm(n) that are not orders of elements of Symm(n-1) (6th row is empty, written as 0 by convention).

%I A080743 #20 Oct 02 2017 10:54:49
%S A080743 1,2,3,4,5,6,0,7,10,12,8,15,9,14,20,21,30,11,18,24,28,35,42,60,13,22,
%T A080743 36,40,33,45,70,84,26,44,56,105,16,39,55,63,66,90,120,140,17,52,72,
%U A080743 210,65,77,78,110,126,132,168,180,19,34,48,88,165,420,51,91,99,130,154,156,220
%N A080743 Array read by rows in which n-th row lists orders of elements of Symm(n) that are not orders of elements of Symm(n-1) (6th row is empty, written as 0 by convention).
%C A080743 A051613 gives number of elements in n-th row.
%H A080743 Alois P. Heinz, <a href="/A080743/b080743.txt">Rows n = 1..100, flattened</a>
%H A080743 J. Bamberg, G. Cairns and D. Kilminster, <a href="http://www.jstor.org/stable/3647934">The crystallographic restriction, permutations and Goldbach's conjecture</a>, Amer. Math. Monthly, 110 (March 2003), 202-209.
%F A080743 n-th row = set of m such that A008475(m) = n, or 0 if no such m exists.
%e A080743 1;
%e A080743 2;
%e A080743 3;
%e A080743 4;
%e A080743 5, 6;
%e A080743 0;
%e A080743 7, 10, 12;
%e A080743 8, 15;
%e A080743 ...
%p A080743 b:= proc(n) option remember; `if`(n<3, {n},
%p A080743       {n, seq(map(x-> ilcm(x, i), b(n-i))[], i=2..n-1)}
%p A080743        minus {seq(b(i)[], i=1..n-1)})
%p A080743     end:
%p A080743 T:= proc(n) local l; l:= [b(n, n)[]];
%p A080743       `if`(nops(l)=0, 0, sort(l)[])
%p A080743     end:
%p A080743 seq(T(n), n=1..20);  # _Alois P. Heinz_, Feb 15 2013
%t A080743 b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0, 1, If[i<1 || n<0, 0, Max[Join[{b[n, i-1]}, Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n]}]]]]] ]; T[1] = {1}; T[6] = {0}; T[n_] := Reap[For[m = n, m <= b[n, PrimePi[n]], m++,  If[n == Total[Power @@@ FactorInteger[m]], Sow[m]]]][[2, 1]]; Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Feb 26 2015, after _Alois P. Heinz_ *)
%Y A080743 Cf. A008475, A051613, A080744, A051703.
%K A080743 nonn,tabf,look,easy
%O A080743 1,2
%A A080743 _N. J. A. Sloane_, Mar 08 2003
%E A080743 More terms from _Vladeta Jovovic_, Mar 12 2003