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A110664
Triangle read by rows: T(n,k)=sum(bigomega(j),j=k..n) (1<=k<=n), where bigomega(j) is the number of prime divisors of j, counted with multiplicities.
1
0, 1, 1, 2, 2, 1, 4, 4, 3, 2, 5, 5, 4, 3, 1, 7, 7, 6, 5, 3, 2, 8, 8, 7, 6, 4, 3, 1, 11, 11, 10, 9, 7, 6, 4, 3, 13, 13, 12, 11, 9, 8, 6, 5, 2, 15, 15, 14, 13, 11, 10, 8, 7, 4, 2, 16, 16, 15, 14, 12, 11, 9, 8, 5, 3, 1, 19, 19, 18, 17, 15, 14, 12, 11, 8, 6, 4, 3, 20, 20, 19, 18, 16, 15, 13, 12, 9, 7
OFFSET
1,4
COMMENTS
T(n,n)=bigomega(n)=A001222(n) =number of prime divisors of n, counted with multiplicities. T(n,1)=sum(bigomega(j),j=1..n) = A022559(n) = sum of exponents in the prime-power factorization of n!.
LINKS
Indranil Ghosh, Rows 1..100, flattened
EXAMPLE
T(4,2)=4 because bigomega(2)+bigomega(3)+bigomega(4)=1+1+2=4.
Triangle begins:
0;
1,1;
2,2,1;
4,4,3,2;
5,5,4,3,1;
MAPLE
with(numtheory): T:=(n, k)->add(bigomega(j), j=k..n): for n from 1 to 13 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
MATHEMATICA
T[n_, n_] := PrimeOmega[n]; T[n_, k_] := Sum[PrimeOmega[j], {j, k, n}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Sep 03 2017 *)
PROG
(PARI) for(n=1, 25, for(k=1, n, print1(sum(j=k, n, bigomega(j)), ", "))) \\ G. C. Greubel, Sep 03 2017
CROSSREFS
Sequence in context: A348840 A182222 A225639 * A193922 A319534 A061436
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Aug 02 2005
STATUS
approved