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A034836 Number of ways to write n as n = x*y*z with 1 <= x <= y <= z. 35
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 9, 4, 2, 1, 10, 2, 2, 2, 6, 1, 10, 2, 4, 2, 2, 2, 12, 1, 4, 4, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Number of boxes with integer edge lengths and volume n.

Starts the same as, but is different from, A033273. First values of n such that a(n) differs from A033273(n) are 36,48,60,64,72,80,84,90,96,100. - Benoit Cloitre, Nov 25 2002

a(n) depends only on the signature of n; the sorted exponents of n. For instance, a(12) and a(18) are the same because both 12 and 18 have signature (1,2). - T. D. Noe, Nov 02 2011

Number of 3D grids of n congruent cubes, in a box, modulo rotation (cf. A007425 and A140773 for boxes instead of cubes; cf. A038548 for the 2D case). - Manfred Boergens, Apr 06 2021

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, 2013.

Index entries for sequences computed from exponents in factorization of n

FORMULA

From Ton Biegstraaten, Jan 04 2016: (Start)

Given a number n, let s(1),...,s(m) be the signature list of n, and a(n) the resulting number in the sequence.

Then np = Product_{k=1..m} binomial(2+s(k),2) is the total number of products solely based on the combination of exponents. The multiplicity of powers is not taken into account (e.g., all combinations of 1,2,4 (6 times) but (2,2,2) only once). See next formulas to compute corrections for 3rd and 2nd powers.

Let ntp = Product_{k=1..m} (floor((s(k) - s(k) mod(3))/s(k))) if the number is a 3rd power or not resulting in 1 or 0.

Let nsq = Product_{k=1..m} (floor(s(k)/2) + 1) is the number of squares.

Conjecture: a(n) = (np + 3*(nsq - ntp) + 5*ntp)/6 = (np + 3*nsq + 2*ntp)/6.

Example: n = 1728; s = [3,6]; np = 10*28 = 280; nsq = 2*4 = 8; ntp = 1 so a(1728)=51 (as in the b-file).

(End)

a(n) >= A226378(n) for all n >= 1. - Antti Karttunen, Aug 30 2017

From Bernard Schott, Dec 12 2021: (Start)

a(n) = 1 iff n = 1 or n is prime (A008578).

a(n) = 2 iff n is semiprime (A001358) (see examples). (End)

EXAMPLE

a(12) = 4 because we can write 12 = 1*1*12 = 1*2*6 = 1*3*4 = 2*2*3.

a(36) = 8 because we can write 36 = 1*1*36 = 1*2*18 = 1*3*12 = 1*4*9 = 1*6*6 = 2*2*9 = 2*3*6 = 3*3*4.

For n = p*q, p < q primes: a(n) = 2 because we can write n = 1*1*pq = 1*p*q.

For n = p^2, p prime: a(n) = 2 because we can write n = 1*1*p^2 = 1*p*p.

MAPLE

f:=proc(n) local t1, i, j, k; t1:=0; for i from 1 to n do for j from i to n do for k from j to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end;

MATHEMATICA

Table[c=0; Do[If[i<=j<=k && i*j*k==n, c++], {i, t=Divisors[n]}, {j, t}, {k, t}]; c, {n, 100}] (* Jayanta Basu, May 23 2013 *)

(* Similar to the first Mathematica code but with fewer steps in Do[..] *)

b=0; d=Divisors[n]; r=Length[d];

Do[If[d[[h]] d[[i]] d[[j]]==n, b++], {h, r}, {i, h, r}, {j, i, r}]; b (* Manfred Boergens, Apr 06 2021 *)

PROG

(PARI) A038548(n)=sumdiv(n, d, d*d<=n) /* <== rhs from A038548 (Michael Somos) */

a(n)=sumdiv(n, d, if(d^3<=n, A038548(n/d) - sumdiv(n/d, d0, d0<d))) \\ Rick L. Shepherd, Aug 27 2006

CROSSREFS

See also: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (this sequence, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered)

Cf. A001358, A008578, A088432, A088433, A088434.

Differs from A033273 and A226378 for the first time at n=36.

Sequence in context: A033273 A319685 A343652 * A316365 A292886 A317508

Adjacent sequences:  A034833 A034834 A034835 * A034837 A034838 A034839

KEYWORD

nonn

AUTHOR

Erich Friedman

EXTENSIONS

Definition simplified by Jonathan Sondow, Oct 03 2013

STATUS

approved

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Last modified May 19 23:40 EDT 2022. Contains 353847 sequences. (Running on oeis4.)