A326028 - OEIS

%I #15 Nov 10 2024 21:47:48

%S 0,1,1,2,1,1,1,2,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,1,

%T 1,5,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2

%N Number of factorizations of n into factors > 1 with integer geometric mean.

%C First differs from A294336 and A316782 at a(36) = 5.

%H Antti Karttunen, <a href="/A326028/b326028.txt">Table of n, a(n) for n = 1..100000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(2^n) = A067538(n).

%e The a(4) = 2 through a(36) = 5 factorizations (showing only the cases where n is a perfect power).

%e   (4)    (8)      (9)    (16)       (25)   (27)     (32)         (36)

%e   (2*2)  (2*2*2)  (3*3)  (2*8)      (5*5)  (3*3*3)  (2*2*2*2*2)  (4*9)

%e                          (4*4)                                   (6*6)

%e                          (2*2*2*2)                               (2*18)

%e                                                                  (3*12)

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],IntegerQ[GeometricMean[#]]&]],{n,2,100}]

%o (PARI) A326028(n, m=n, facmul=1, facnum=0) = if(1==n,facnum>0 && ispower(facmul,facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326028(n/d, d, facmul*d, facnum+1))); (s)); \\ _Antti Karttunen_, Nov 10 2024

%Y Positions of terms > 1 are the perfect powers A001597.

%Y Partitions with integer geometric mean are A067539.

%Y Subsets with integer geometric mean are A326027.

%Y Factorizations with integer average and geometric mean are A326647.

%Y Cf. A001055, A082553, A322794, A326514, A326515, A326516, A326622, A326623, A326624, A326625.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jul 15 2019

%E a(89) onwards from _Antti Karttunen_, Nov 10 2024