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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.

%I A082786 #57 Feb 16 2025 08:32:49
%S A082786 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,
%T A082786 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,
%U A082786 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
%N 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.
%C A082786 n = Product_{k=1..n} prime(k)^T(n,k);
%C A082786 T(n, A055396(n)) > 0 and T(n,k) = 0 for 1 <= k < A055396(n);
%C A082786 T(n, A061395(n)) > 0 and T(n,k) = 0 for A061395(n) < k <= n;
%C A082786 Sum_{k=1..n} T(n,k) = A001222(n);
%C A082786 Sum_{k=1..n} A057427(T(n,k)) = A001221(n);
%C A082786 Sum_{k=1..n} T(n,k)*prime(k) = A001414(n);
%C A082786 Sum_{k=1..n} A057427(T(n,k))*prime(k) = A008472(n);
%C A082786 Min(T(n,k): 1<=k<=n) = A051904(n);
%C A082786 Max(T(n,k): 1<=k<=n) = A051903(n);
%C A082786 T(n,1) = A007814(n); T(n,2) = A007949(n), n>1.
%H A082786 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactorization.html">Prime Factorization</a>
%e A082786 Triangle begins:
%e A082786   0,
%e A082786   1, 0,
%e A082786   0, 1, 0,
%e A082786   2, 0, 0, 0,
%e A082786   0, 0, 1, 0, 0,
%e A082786   1, 1, 0, 0, 0, 0,
%e A082786   0, 0, 0, 1, 0, 0, 0,
%e A082786   3, 0, 0, 0, 0, 0, 0, 0,
%e A082786   ...
%t A082786 Table[IntegerExponent[n, Prime[k]], {n,1,15}, {k,1,n}] // Flatten  (* _Amiram Eldar_, Dec 14 2018 *)
%o A082786 (PARI) row(n) = vector(n, k, valuation(n, prime(k)));
%o A082786 tabl(nn) = for (n=1, nn, print(row(n))); \\ _Michel Marcus_, Dec 14 2018
%Y A082786 Cf. A000040, A049084.
%Y A082786 Cf. A067255 (same as irregular triangle).
%K A082786 nonn,tabl
%O A082786 1,7
%A A082786 _Reinhard Zumkeller_, May 22 2003