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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 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. From Reinhard Zumkeller, Oct 01 2012: (Start) 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. (End) 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=(d for d in divisors(n)[1:-1] if d <= m)     s=sum(T(n//d, d) for d in A)     return s + 1 if n<=m else s for n in range(1, 21): print([T(n, m) for m in range(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 December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)