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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.

%I A143078 #24 May 01 2013 21:02:10
%S A143078 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,
%T A143078 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,
%U A143078 1,2,0,0,0,0,0,0,0,0,0,0,0,0,1,2,0,1
%N 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.
%C A143078 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
%C A143078 Otherwise said, the number of suppressed (= trailing) 0's in row n is A000720(n)-A061395(n). - _M. F. Hasler_, Mar 10 2013
%F A143078 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).
%e A143078 Triangle begins
%e A143078 {1},
%e A143078 {0, 1},
%e A143078 {2, 0},
%e A143078 {0, 0, 1},
%e A143078 {1, 1, 0}, (the 6th row, and 6 = prime(1)*prime(2))
%e A143078 {0, 0, 0, 1},
%e A143078 {3, 0, 0, 0},
%e A143078 {0, 2, 0, 0},
%e A143078 {1, 0, 1, 0},
%e A143078 ...
%t A143078 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[%]
%o A143078 (PARI) my(r(n)=vector(primepi(n),i,valuation(n,prime(i)))); concat(vector(20,n,r(n))) \\ [_M. F. Hasler_, Mar 10 2013]
%Y A143078 Cf. A000720, A001222, A067255.
%K A143078 nonn,tabf,easy
%O A143078 2,4
%A A143078 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 14 2008
%E A143078 Edited by _N. J. A. Sloane_, Jan 12 2012
%E A143078 More terms from _M. F. Hasler_, Mar 10 2013