A124960 -id:A124960 - cp's OEIS frontend

A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 5, 4, 0, 1, 10, 19, 8, 0, 1, 17, 69, 65, 16, 0, 1, 28, 188, 410, 211, 32, 0, 1, 41, 496, 1726, 2261, 665, 64, 0, 1, 58, 1029, 7182, 14343, 11970, 2059, 128, 0, 1, 77, 2015, 20559, 93345, 112371, 61909, 6305, 256, 0, 1, 100, 3478, 54814, 360612, 1139166, 848506, 315850, 19171, 512, 0

Offset: 0

Alois P. Heinz, Apr 27 2021

			A(1,3) = 10 = 5 + 3 + 2, sum of encoded compositions [0,0,1], [0,1,0], [1,0,0].
A(4,2) = 211 = 81 + 54 + 36 + 24 + 16, sum of encoded compositions [0,4], [1,3], [2,2], [3,1], [4,0].
Square array A(n,k) begins:
  1,  1,    1,     1,      1,        1,        1, ...
  0,  2,    5,    10,     17,       28,       41, ...
  0,  4,   19,    69,    188,      496,     1029, ...
  0,  8,   65,   410,   1726,     7182,    20559, ...
  0, 16,  211,  2261,  14343,    93345,   360612, ...
  0, 32,  665, 11970, 112371,  1139166,  5827122, ...
  0, 64, 2059, 61909, 848506, 13379332, 89131918, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-4 give: A000007, A000079, A001047(n+1), A016273, A025931.
Rows n=0-2 give: A000012, A007504, A357251.
Main diagonal gives A332967.
Cf. A000040, A074140, A124960, A324939, A325054.

A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, add(ithprime(k)^i*A(n-i, k-1), i=0..n)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
     `if`(k=0, 0, ithprime(k)*A(n-1, k)+A(n, k-1)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1,
     If[k == 0, 0, Prime[k] A[n-1, k] + A[n, k-1]]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 2nd Maple program *)

A(n,k) = [x^n] Product_{i=1..k} 1/(1-prime(i)*x).

A(n,k) = A124960(n+k,k) for k >= 1.

A332967 Sum of all integers m satisfying Omega(m) = n and pi(p) <= n for all prime factors p of m.

Original entry on oeis.org

1, 2, 19, 410, 14343, 1139166, 89131918, 10861230692, 1271028562739, 203393524967230, 52274418436233714, 11160490802017899420, 3415612116240107778630, 1088775430914588654276060, 311608007930071575510930780, 99738699420765496000734958440

Offset: 0

Alois P. Heinz, Mar 04 2020

nonn

			a(2) = 4 + 6 + 9 = 2*2 + 2*3 + 3*3 = 19.

Alois P. Heinz, Table of n, a(n) for n = 0..279

Row sums of A330394.
Main diagonal of A343751.
Cf. A000040, A000720, A001222, A027748, A088218, A124960.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(ithprime(j)*b(n-1, j), j=1..i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);

a(n) = [x^n] Product_{i=1..n} 1/(1-prime(i)*x).
a(n) = A124960(2n,n).
a(n) = Sum_{k=1..A088218(n)} A330394(n,k).
a(n) = A343751(n,n).

A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.

Original entry on oeis.org

1, 2, 4, 6, 9, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 2401, 32, 48, 72, 80, 108, 112, 120, 162

Offset: 0

Robert Price, Mar 03 2020

Positive integers not in T are: 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, ... .
Row n has exactly one squarefree member: primorial(n) = A002110(n).
Sorting all terms (except 1) gives A324521.

			Triangle T(n,k) begins:
  1;
  2;
  4,  6,  9;
  8, 12, 18, 20, 27, 30, 45, 50, 75, 125;
  ...

Robert Price, Table of n, a(n) for n = 0..8788

Column k=1 gives A000079.
Last elements of rows give A307539.
Row lengths give A088218.
Row sums give A332967(n) = A124960(2n,n).
T(n,n) gives A101695(n) for n > 0.
Cf. A000720, A001222, A005117, A027748, A324521.

b:= proc(n, i) option remember; `if`(n=0, [1], [seq(
      map(x-> x*ithprime(j), b(n-1, j))[], j=1..i)])
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Mar 03 2020

t = Table[Union[Apply[Times, Tuples[Prime[Range[n]], {n}], {1}]], {n, 0, 5}];
t // TableForm
Flatten[t]

cp's OEIS Frontend

A343751 A(n,k) is the sum of all compositions [c_1, c_2, ..., c_k] of n into k nonnegative parts encoded as Product_{i=1..k} prime(i)^(c_i); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A332967 Sum of all integers m satisfying Omega(m) = n and pi(p) <= n for all prime factors p of m.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica