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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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);

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 A257900

Adjacent sequences:  A333362 A333363 A333364 * A333366 A333367 A333368

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Mar 16 2020

STATUS

approved

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Last modified May 31 22:04 EDT 2020. Contains 334756 sequences. (Running on oeis4.)