A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

1, 1, 2, 1, 2, 3, 1, 3, 4, 4, 1, 3, 4, 4, 5, 1, 3, 5, 5, 6, 6, 1, 3, 5, 5, 6, 6, 7, 1, 4, 6, 6, 7, 7, 8, 8, 1, 4, 7, 7, 8, 8, 9, 9, 9, 1, 4, 7, 7, 9, 9, 10, 10, 10, 10, 1, 4, 7, 7, 9, 9, 10, 10, 10, 10, 11, 1, 4, 8, 8, 10, 10, 11, 11, 11, 11, 12, 12, 1, 4, 8, 8, 10, 10, 11, 11, 11, 11, 12, 12, 13, 1, 4

Offset

1,3

Comments

T(n,n-1) = A014684(n) for n>1;

T(n,2) = A029837(n) for n>1; T(n,3) = A071521(n) for n>2; T(n,5) = A071520(n) for n>4.

A036234(n) = number of distinct terms in n-th row. - Reinhard Zumkeller, Sep 17 2013

Links

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened

Eric Weisstein's World of Mathematics, Smooth Number.

Example

Triangle begins:
.................. 1
................ 1...2
.............. 1...2...3
............ 1...3...4...4
.......... 1...3...4...4...5
........ 1...3...5...5...6...6
...... 1...3...5...5...6...6...7
.... 1...4...6...6...7...8...8...8
.. 1...4...7...7...8...8...9...9...9.

Maple

A080786 := proc(x, y)
    local a, n ;
    a := 0 ;
    for n from 1 to x do
        if A006530(n) <= y then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 31 2013

Mathematica

P[n_] := FactorInteger[n][[-1, 1]]; P[1]=1; T[n_, k_] := (For[j=0; m=1, m <= n, m++, If[P[m] <= k, j++]]; j); Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2015 *)

Prog

(Haskell)
a080786 n k = a080786_tabl !! (n-1) !! (k-1)
a080786_row n = a080786_tabl !! (n-1)
a080786_tabl = map reverse $ iterate f [1] where
   f xs@(x:_) = (x + 1) :
                (zipWith (+) xs (map (fromEnum . (lpf <=)) [x, x-1 ..]))
        where lpf = fromInteger $ a006530 $ fromIntegral (x + 1)
-- Reinhard Zumkeller, Sep 17 2013

Crossrefs

Cf. A000079, A002473, A003586, A006530, A014684, A029837, A036234, A051037, A051038, A071520, A071521, A080197, A080681, A080682, A080683.

Sequence in context: A344088 A292595 A269596 * A036838 A066010 A209556

Adjacent sequences: A080783 A080784 A080785 * A080787 A080788 A080789

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Mar 12 2003

Status

approved