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A326622 Number of factorizations of n into factors > 1 with integer average.

%I A326622 #15 Nov 10 2024 21:48:01
%S A326622 0,1,1,2,1,1,1,3,2,1,1,2,1,1,2,4,1,1,1,3,2,1,1,4,2,1,3,2,1,1,1,5,2,1,
%T A326622 2,3,1,1,2,3,1,2,1,3,3,1,1,6,2,2,2,2,1,2,2,4,2,1,1,6,1,1,3,7,2,1,1,3,
%U A326622 2,1,1,6,1,1,3,2,2,2,1,7,5,1,1,4,2,1,2,3,1,1,2,3,2,1,2,8,1,1,3,3,1,1,1,4,5,1,1,6
%N A326622 Number of factorizations of n into factors > 1 with integer average.
%H A326622 Antti Karttunen, <a href="/A326622/b326622.txt">Table of n, a(n) for n = 1..65537</a>
%e A326622 The a(80) = 7 factorizations:
%e A326622   (2*2*2*10)
%e A326622   (2*2*20)
%e A326622   (2*5*8)
%e A326622   (2*40)
%e A326622   (4*20)
%e A326622   (8*10)
%e A326622   (80)
%t A326622 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A326622 Table[Length[Select[facs[n],IntegerQ[Mean[#]]&]],{n,2,100}]
%o A326622 (PARI) A326622(n, m=n, facsum=0, facnum=0) = if(1==n,facnum > 0 && 1==denominator(facsum/facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326622(n/d, d, facsum+d, facnum+1))); (s)); \\ _Antti Karttunen_, Nov 10 2024
%Y A326622 Partitions with integer average are A067538.
%Y A326622 Strict partitions with integer average are A102627.
%Y A326622 Heinz numbers of partitions with integer average are A316413.
%Y A326622 Factorizations with integer geometric mean are A326028.
%Y A326622 Cf. A001055, A051293, A078174, A078175, A326514, A326515, A326567/A326568, A326621, A326623, A326667 (= a(2^n)), A327906 (positions of 1's), A327907 (of terms > 1).
%K A326622 nonn
%O A326622 1,4
%A A326622 _Gus Wiseman_, Jul 14 2019
%E A326622 Data section extended up to a(108), with missing term a(1)=0 also added (thus correcting the offset) - _Antti Karttunen_, Nov 10 2024