login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)