A143078 Triangle read by rows: row n (n >= 2) has length pi(n) (see A000720) and the k-th term gives the exponent of prime(k) in the prime factorization of n.

1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1

Offset: 2

Author

Roger L. Bagula and Gary W. Adamson, Oct 14 2008

Keywords

• nonn
• tabf
• easy

Comments

If we suppress the 0's at the ends of the rows we get A067255. The number of 0's suppressed is A036234(n)-A061395(n)-1. - Jacques ALARDET, Jan 11 2012

Otherwise said, the number of suppressed (= trailing) 0's in row n is A000720(n)-A061395(n). - M. F. Hasler, Mar 10 2013

Examples

			Triangle begins
{1},
{0, 1},
{2, 0},
{0, 0, 1},
{1, 1, 0}, (the 6th row, and 6 = prime(1)*prime(2))
{0, 0, 0, 1},
{3, 0, 0, 0},
{0, 2, 0, 0},
{1, 0, 1, 0},
...
		

Crossrefs

Cf. A000720, A001222, A067255.

Programs

Mathematica

Clear[t, T, n, m, k]; t[n_, m_, k_] := If[PrimeQ[FactorInteger[ n][[m]][[1]]] && FactorInteger[n][[m]][[1]] == Prime[k], FactorInteger[n][[m]][[2]], 0]; T = Table[Apply[Plus, Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], { m, 1, Length[FactorInteger[n]]}]], {n, 1, 10}]; Flatten[%]

PARI

my(r(n)=vector(primepi(n),i,valuation(n,prime(i)))); concat(vector(20,n,r(n))) \\ [M. F. Hasler, Mar 10 2013]

Formula

t(n,m,k)=If[PrimeQ[FactorInteger[n][[m]][[1]]] && FactorInteger[n][[m]][[ 1]] == Prime[k], FactorInteger[n][[m]][[2]], 0]; T(n,m)=vector_sum overk of t(n,m,k).

Extensions

Edited by N. J. A. Sloane, Jan 12 2012

More terms from M. F. Hasler, Mar 10 2013