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A034836 Number of ways to write n as n = x*y*z with 1 <= x <= y <= z.

%I A034836 #79 Oct 25 2024 09:48:56
%S A034836 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,
%T A034836 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,
%U A034836 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
%N A034836 Number of ways to write n as n = x*y*z with 1 <= x <= y <= z.
%C A034836 Number of boxes with integer edge lengths and volume n.
%C A034836 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
%C A034836 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
%C A034836 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
%H A034836 T. D. Noe, <a href="/A034836/b034836.txt">Table of n, a(n) for n = 1..10000</a>
%H A034836 Dorin Andrica and Eugen J. Ionascu, <a href="https://doi.org/10.2478/auom-2014-0001">On the number of polynomials with coefficients in [n]</a>, An. Şt. Univ. Ovidius Constanţa, Vol. 22, No. 1 (2013), pp. 13-23; <a href="http://www.emis.de/journals/ASUO/mathematics_/vol22-1/Andrica_D__Ionascu_E.J._nou-1__final_.pdf">alternative link</a>.
%H A034836 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A034836 From _Ton Biegstraaten_, Jan 04 2016: (Start)
%F A034836 Given a number n, let s(1),...,s(m) be the signature list of n, and a(n) the resulting number in the sequence.
%F A034836 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.
%F A034836 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.
%F A034836 Let nsq = Product_{k=1..m} (floor(s(k)/2) + 1) is the number of squares.
%F A034836 Conjecture: a(n) = (np + 3*(nsq - ntp) + 5*ntp)/6 = (np + 3*nsq + 2*ntp)/6.
%F A034836 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).
%F A034836 (End)
%F A034836 a(n) >= A226378(n) for all n >= 1. - _Antti Karttunen_, Aug 30 2017
%F A034836 From _Bernard Schott_, Dec 12 2021: (Start)
%F A034836 a(n) = 1 iff n = 1 or n is prime (A008578).
%F A034836 a(n) = 2 iff n is semiprime (A001358) (see examples). (End)
%F A034836 a(n) = (2 * A010057(n) + A007425(n) + 3 * A046951(n))/6 (Andrica and Ionascu, 2013, p. 19, eq. 11). - _Amiram Eldar_, Apr 19 2024
%e A034836 a(12) = 4 because we can write 12 = 1*1*12 = 1*2*6 = 1*3*4 = 2*2*3.
%e A034836 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.
%e A034836 For n = p*q, p < q primes: a(n) = 2 because we can write n = 1*1*pq = 1*p*q.
%e A034836 For n = p^2, p prime: a(n) = 2 because we can write n = 1*1*p^2 = 1*p*p.
%p A034836 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;
%p A034836 # second Maple program:
%p A034836 A034836:=proc(n)
%p A034836    local a,b,i;
%p A034836    a:=0;
%p A034836    b:=(l,x,h)->l<=x and x<=h;
%p A034836    for i in select(`<=`,NumberTheory:-Divisors(n),iroot(n,3)) do
%p A034836       a:=a+nops(select[2](b,i,NumberTheory:-Divisors(n/i),isqrt(n/i)))
%p A034836    od;
%p A034836    return a
%p A034836 end proc;
%p A034836 seq(A034836(n),n=1..100); # _Felix Huber_, Oct 02 2024
%t A034836 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 *)
%t A034836 (* Similar to the first Mathematica code but with fewer steps in Do[..] *)
%t A034836 b=0; d=Divisors[n]; r=Length[d];
%t A034836 Do[If[d[[h]] d[[i]] d[[j]]==n, b++], {h, r}, {i, h, r}, {j, i, r}]; b (* _Manfred Boergens_, Apr 06 2021 *)
%t A034836 a[1] = 1; a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[IntegerQ[Surd[n, 3]], 1/3, 0] + (Times @@ ((e + 1)*(e + 2)/2))/6 + (Times @@ (Floor[e/2] + 1))/2]; Array[a, 100] (* _Amiram Eldar_, Apr 19 2024 *)
%o A034836 (PARI) A038548(n)=sumdiv(n, d, d*d<=n) /* <== rhs from A038548 (_Michael Somos_) */
%o A034836 a(n)=sumdiv(n, d, if(d^3<=n, A038548(n/d) - sumdiv(n/d, d0, d0<d))) \\ _Rick L. Shepherd_, Aug 27 2006
%o A034836 (PARI) a(n) = {my(e = factor(n)[,2]); (2 * ispower(n, 3) + vecprod(apply(x -> (x+1)*(x+2)/2, e)) + 3 * vecprod(apply(x -> x\2 + 1, e))) / 6;} \\ _Amiram Eldar_, Apr 19 2024
%Y A034836 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)
%Y A034836 Cf. A001358, A008578, A010057, A046951, A088432, A088433, A088434.
%Y A034836 Differs from A033273 and A226378 for the first time at n=36.
%K A034836 nonn
%O A034836 1,4
%A A034836 _Erich Friedman_
%E A034836 Definition simplified by _Jonathan Sondow_, Oct 03 2013