A333365 T(n,k) is the number of times that prime(k) is the least part in a partition of n into prime parts; triangle T(n,k), n >= 0, 1 <= k <= max(1,A000720(A331634(n))), read by rows.

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 0, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 0, 0, 1, 5, 1, 1, 6, 2, 0, 0, 0, 1, 7, 2, 0, 1, 9, 2, 1, 10, 3, 1, 12, 3, 1, 0, 0, 0, 1, 14, 3, 1, 1, 17, 4, 1, 0, 0, 0, 0, 1, 19, 5, 1, 1, 23, 5, 1, 1, 26, 6, 2, 0, 1, 30, 7, 2, 0, 0, 0, 0, 0, 1

Offset

0,12

Links

Alois P. Heinz, Rows n = 0..1000, flattened

Formula

T(n,pi(n)) = A010051(n) for n > 1.

T(p,pi(p)) = 1 if p is prime.

T(prime(k),k) = 1 for k >= 1.

Recursion: T(n,k) = Sum_{q=k..pi(n-p)} T(n-p, q) with p := prime(k) and T(n,k) = 0 if n < p, or 1 if n = p. - David James Sycamore, Mar 28 2020

Example

In the A000607(11) = 6 partitions of 11 into prime parts, (11), 335, 227, 2225, 2333, 22223 the least parts are 11 = prime(5) (once), 3 = prime(2)(once), and 2 = prime(1) (four times), whereas 5 and 7 (prime(3) and prime(4)) do not occur. Thus row 11 is [4,1,0,0,1].
Triangle T(n,k) begins:
   0    ;
   0    ;
   1    ;
   0, 1    ;
   1       ;
   1, 0, 1    ;
   1, 1       ;
   2, 0, 0, 1    ;
   2, 1          ;
   3, 1          ;
   3, 1, 1       ;
   4, 1, 0, 0, 1    ;
   5, 1, 1          ;
   6, 2, 0, 0, 0, 1    ;
   7, 2, 0, 1          ;
   9, 2, 1             ;
  10, 3, 1             ;
  12, 3, 1, 0, 0, 0, 1    ;
  14, 3, 1, 1             ;
  17, 4, 1, 0, 0, 0, 0, 1    ;
  19, 5, 1, 1                ;
  ...

Maple

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i),
      coeff(p, x, i), [][]), i=2..max(2, degree(p))))(b(n, 2, x))
    end:
seq(T(n), n=0..23);

Mathematica

b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, With[{q = NextPrime[p]}, Sum[b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]]]];
T[n_] := If[n < 2, {0}, MapIndexed[If[PrimeQ[#2[[1]]], #1, Nothing]&, Rest @ CoefficientList[b[n, 2, x], x]]];
T /@ Range[0, 23] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)

Crossrefs

Columns k=1-2 give: A000607(n-2) for n>1, A099773(n-3) for n>2.

Row sums give A000607 for n>0.

Length of n-th row is A000720(A331634(n)) for n>1.

Indices of rows without 1's: A330433.

Cf. A000040, A000720, A010051, A333129, A333238, A333259.

Sequence in context: A292598 A079113 A144874 * A303065 A325406 A360675

Adjacent sequences: A333362 A333363 A333364 * A333366 A333367 A333368

Keyword

nonn,tabf

Author

Alois P. Heinz, Mar 16 2020

Status

approved