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A066032 Number of ways to write n as a product with no factor larger than m (1 <= m <=n, written row by row). 8
1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 2, 2, 3, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

T(n,floor(n/2)) = A028422(n) for n > 1; T(n,floor(n/3)) = A216599(n) for n > 2; T(n,floor(n/4)) = A216600(n) for n > 3; T(n,floor(n/5)) = A216601(n) for n > 4; T(n,floor(n/6)) = A216602(n) for n > 5. - Reinhard Zumkeller, Oct 01 2012

LINKS

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

FORMULA

T(1, 1) = 1. For every prime p T(p, m) = 1 if p <= m and 0 else. For composite n: T(n, m) = sum[T(n/d, d)] + I(n<=m) where the sum is over all divisors d of n except 1 and n with d <= m and I(n<=m) is 1 if n<=m and 0 else.

EXAMPLE

T(12, 5) = a(71) = 2, since there are 2 possibilities to write 12 as a product with no factor larger than 5 (4*3 and 3*2*2)

1;

0,1;

0,0,1;

0,1,1,2;

0,0,0,0,1;

0,0,1,1,1,2;

0,0,0,0,0,0,1;

0,1,1,2,2,2,2,3;

0,0,1,1,1,1,1,1,2;

0,0,0,0,1,1,1,1,1,2;

0,0,0,0,0,0,0,0,0,0,1;

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

MAPLE

with(numtheory): T := proc(n::integer, m::integer) local i, A, summe, d: if isprime(n) then: if n <= m then RETURN(1) fi: RETURN(0): fi:

A := divisors(n) minus {n, 1}: for d in A do: if d > m then A := A minus {d}: fi: od: summe := 0: for d in A do: summe := summe + T(n/d, d): od: if n <=m then summe := summe + 1: fi: RETURN(summe): end: A066032 := [seq(seq(T(n, m), m=1..n), n=1..16)];

MATHEMATICA

T[1, 1] = 1; T[p_?PrimeQ, m_] := Boole[p <= m]; T[n_, m_] := Sum[T[n/d, d]* Boole[d <= m], {d, Divisors[n][[2 ;; -2]]}] + Boole[n <= m];

Table[T[n, m], {n, 1, 14}, {m, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

PROG

(Haskell)

a066032 1 1 = 1

a066032 n k = fromEnum (n <= k) +

   (sum $ map (\d -> a066032 (n `div` d) d) $

              takeWhile (<= k) $ tail $ a027751_row n)

a066032_row n = map (a066032 n) [1..n]

a066032_tabl = map a066032_row [1..]

-- Reinhard Zumkeller, Oct 01 2012

(Python)

from sympy import divisors, isprime

def T(n, m):

    if isprime(n): return 1 if n<=m else 0

    A=filter(lambda d: d<=m, divisors(n)[1:-1])

    s=sum([T(n/d, d) for d in A])

    return s + 1 if n<=m else s

for n in xrange(1, 21): print [T(n, m) for m in xrange(1, n + 1)] # Indranil Ghosh, Aug 19 2017

CROSSREFS

A001055(n) = T(n, n) is the right diagonal.

Sequence in context: A127475 A086014 A025437 * A035187 A291147 A278929

Adjacent sequences:  A066029 A066030 A066031 * A066033 A066034 A066035

KEYWORD

nonn,look,tabl

AUTHOR

Ulrich Schimke (ulrschimke(AT)aol.com), Feb 11 2002

STATUS

approved

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Last modified March 21 12:12 EDT 2019. Contains 321369 sequences. (Running on oeis4.)