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A317390
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.
14
2, 1, 5, 25, 3, 7, 43, 29, 4, 11, 211, 61, 37, 6, 15, 638, 261, 91, 40, 8, 23, 664, 848, 421, 111, 41, 9, 26, 1613, 1956, 921, 426, 121, 49, 10, 27, 2991, 3321, 2058, 969, 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
OFFSET
1,1
FORMULA
A317241(A(n,k)) = k.
EXAMPLE
A(6,2) = 49: 1 + 3 * (1 + 5 * (1 + 2)) = 1 + 2 * (1 + 23) = 49.
Square array A(n,k) begins:
2, 1, 25, 43, 211, 638, 664, 1613, 2991, ...
5, 3, 29, 61, 261, 848, 1956, 3321, 3004, ...
7, 4, 37, 91, 421, 921, 2058, 3336, 3319, ...
11, 6, 40, 111, 426, 969, 2092, 3368, 3554, ...
15, 8, 41, 121, 441, 1002, 2094, 3741, 3928, ...
23, 9, 49, 124, 484, 1026, 2283, 3914, 4846, ...
26, 10, 51, 171, 535, 1106, 2381, 3979, 5552, ...
27, 12, 52, 184, 540, 1156, 2388, 4082, 5886, ...
28, 13, 67, 187, 591, 1191, 2432, 4126, 6293, ...
MAPLE
b:= proc(n, s) option remember; `if`(n=1, 1, add(b((n-1)/p,
s union {p}) , p=numtheory[factorset](n-1) minus s))
end:
A:= proc() local h, p, q; p, q:= proc() [] end, 0;
proc(n, k)
while nops(p(k))<n do q:= q+1;
h:= b(q, {});
p(h):= [p(h)[], q]
od; p(k)[n]
end
end():
seq(seq(A(n, d-n), n=1..d), d=1..10);
MATHEMATICA
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}]];
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]]];
Table[Table[A[n, d - n], {n, 1, d}], {d, 1, 11}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)
CROSSREFS
Row n=1 gives A317385.
A(n,n) gives A317537.
Cf. A317241.
Sequence in context: A349852 A260701 A184300 * A370163 A075403 A260503
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 27 2018
STATUS
approved