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

2,4

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

Links

Table of n, a(n) for n=2..87.

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).

Example

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},
...

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[%]

Prog

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

Crossrefs

Cf. A000720, A001222, A067255.

Sequence in context: A276675 A236389 A337821 * A106405 A228601 A363856

Adjacent sequences: A143075 A143076 A143077 * A143079 A143080 A143081

Keyword

nonn,tabf,easy

Author

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

Extensions

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

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

Status

approved