A082786 Triangle, read by rows, of exponents of primes in canonical prime factorization of n: T(n,k) = greatest number such that prime(k)^T(n,k) divides n, 1 <= k <= n.

0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, May 22 2003

Keywords

• nonn
• tabl

Comments

n = Product_{k=1..n} prime(k)^T(n,k);

T(n, A055396(n)) > 0 and T(n,k) = 0 for 1 <= k < A055396(n);

T(n, A061395(n)) > 0 and T(n,k) = 0 for A061395(n) < k <= n;

Sum_{k=1..n} T(n,k) = A001222(n);

Sum_{k=1..n} A057427(T(n,k)) = A001221(n);

Sum_{k=1..n} T(n,k)*prime(k) = A001414(n);

Sum_{k=1..n} A057427(T(n,k))*prime(k) = A008472(n);

Min(T(n,k): 1<=k<=n) = A051904(n);

Max(T(n,k): 1<=k<=n) = A051903(n);

T(n,1) = A007814(n); T(n,2) = A007949(n), n>1.

Examples

			Triangle begins:
  0,
  1, 0,
  0, 1, 0,
  2, 0, 0, 0,
  0, 0, 1, 0, 0,
  1, 1, 0, 0, 0, 0,
  0, 0, 0, 1, 0, 0, 0,
  3, 0, 0, 0, 0, 0, 0, 0,
  ...
		

Links

• Eric Weisstein's World of Mathematics, Prime Factorization

Crossrefs

Cf. A000040, A049084.

Cf. A067255 (same as irregular triangle).

Programs

Mathematica

Table[IntegerExponent[n, Prime[k]], {n,1,15}, {k,1,n}] // Flatten  (* Amiram Eldar, Dec 14 2018 *)

PARI

row(n) = vector(n, k, valuation(n, prime(k)));
tabl(nn) = for (n=1, nn, print(row(n))); \\ Michel Marcus, Dec 14 2018