A034836 Number of ways to write n as n = x*y*z with 1 <= x <= y <= z.
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
Offset: 1
Keywords
Examples
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.
Links
- T. D. Noe, Table of n, a(n) for n = 1..10000
- Dorin Andrica and Eugen J. Ionascu, On the number of polynomials with coefficients in [n], An. Şt. Univ. Ovidius Constanţa, Vol. 22, No. 1 (2013), pp. 13-23; alternative link.
- Index entries for sequences computed from exponents in factorization of n.
Crossrefs
Programs
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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; # second Maple program: A034836:=proc(n) local a,b,i; a:=0; b:=(l,x,h)->l<=x and x<=h; for i in select(`<=`,NumberTheory:-Divisors(n),iroot(n,3)) do a:=a+nops(select[2](b,i,NumberTheory:-Divisors(n/i),isqrt(n/i))) od; return a end proc; seq(A034836(n),n=1..100); # Felix Huber, Oct 02 2024
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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 *) 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 *) -
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
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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
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)
a(n) = (2 * A010057(n) + A007425(n) + 3 * A046951(n))/6 (Andrica and Ionascu, 2013, p. 19, eq. 11). - Amiram Eldar, Apr 19 2024
Extensions
Definition simplified by Jonathan Sondow, Oct 03 2013
Comments