A317390 - OEIS

%I #22 Dec 06 2019 08:53:27

%S 2,1,5,25,3,7,43,29,4,11,211,61,37,6,15,638,261,91,40,8,23,664,848,

%T 421,111,41,9,26,1613,1956,921,426,121,49,10,27,2991,3321,2058,969,

%U 441,124,51,12,28,7021,3004,3336,2092,1002,484,171,52,13,31,11306,7162,3319,3368,2094,1026,535,184,67,14,33

%N A(n,k) is the n-th positive integer that has exactly k representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes; square array A(n,k), n>=1, k>=0, read by antidiagonals.

%H Alois P. Heinz, <a href="/A317390/b317390.txt">Antidiagonals n = 1..34, flattened</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F A317241(A(n,k)) = k.

%e A(6,2) = 49: 1 + 3 * (1 + 5 * (1 + 2)) = 1 + 2 * (1 + 23) = 49.

%e Square array A(n,k) begins:

%e    2,  1, 25,  43, 211,  638,  664, 1613, 2991, ...

%e    5,  3, 29,  61, 261,  848, 1956, 3321, 3004, ...

%e    7,  4, 37,  91, 421,  921, 2058, 3336, 3319, ...

%e   11,  6, 40, 111, 426,  969, 2092, 3368, 3554, ...

%e   15,  8, 41, 121, 441, 1002, 2094, 3741, 3928, ...

%e   23,  9, 49, 124, 484, 1026, 2283, 3914, 4846, ...

%e   26, 10, 51, 171, 535, 1106, 2381, 3979, 5552, ...

%e   27, 12, 52, 184, 540, 1156, 2388, 4082, 5886, ...

%e   28, 13, 67, 187, 591, 1191, 2432, 4126, 6293, ...

%p b:= proc(n, s) option remember; `if`(n=1, 1, add(b((n-1)/p,

%p       s union {p}) , p=numtheory[factorset](n-1) minus s))

%p     end:

%p A:= proc() local h, p, q; p, q:= proc() [] end, 0;

%p       proc(n, k)

%p         while nops(p(k))<n do q:= q+1;

%p           h:= b(q, {});

%p           p(h):= [p(h)[], q]

%p         od; p(k)[n]

%p       end

%p     end():

%p seq(seq(A(n, d-n), n=1..d), d=1..10);

%t b[n_, s_List] := b[n, s] = If[n == 1, 1, Sum[If[p == 1, 0, b[(n - 1)/p, s  ~Union~ {p}]], {p, FactorInteger[n - 1][[All, 1]] ~Complement~ s}]];

%t A[n_, k_] := Module[{h, p, q = 0}, p[_] = {}; While[Length[p[k]] < n, q = q + 1; h = b[q, {}]; p[h] = Append[p[h], q]]; p[k][[n]]];

%t Table[Table[A[n, d - n], {n, 1, d}], {d, 1, 11}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, from Maple *)

%Y Columns k=0-10 give: A317242, A317391, A317392, A317393, A317394, A317395, A317396, A317397, A317398, A317399, A317400.

%Y Row n=1 gives A317385.

%Y A(n,n) gives A317537.

%Y Cf. A317241.

%K nonn,tabl

%O 1,1

%A _Alois P. Heinz_, Jul 27 2018