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A327906 Numbers with only one factorization into factors > 1 with integer mean (namely, as a singleton).

%I A327906 #9 Nov 10 2024 21:47:53
%S A327906 2,3,5,6,7,10,11,13,14,17,18,19,22,23,26,29,30,31,34,37,38,41,43,46,
%T A327906 47,53,58,59,61,62,66,67,70,71,73,74,79,82,83,86,89,90,94,97,98,101,
%U A327906 102,103,106,107,109,113,118,122,127,130,131,134,137,138,139,142
%N A327906 Numbers with only one factorization into factors > 1 with integer mean (namely, as a singleton).
%H A327906 Antti Karttunen, <a href="/A327906/b327906.txt">Table of n, a(n) for n = 1..10000</a>
%e A327906 There are 4 factorizations of 24 with integer mean, namely:
%e A327906   (24)
%e A327906   (4*6)
%e A327906   (2*12)
%e A327906   (2*3*4)
%e A327906 so 24 is not in the sequence.
%t A327906 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A327906 Select[Range[100],Length[Select[facs[#],IntegerQ[Mean[#]]&]]==1&]
%o A327906 (PARI)
%o A327906 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
%o A327906 isA327906(n) = (1==A326622(n)); \\ _Antti Karttunen_, Nov 10 2024
%Y A327906 Complement of A327907.
%Y A327906 Positions of 1's in A326622.
%Y A327906 Cf. A001055, A051293, A067538, A078175, A123528/A123529, A316413, A326515, A326516, A326666, A326667, A326671.
%K A327906 nonn
%O A327906 1,1
%A A327906 _Gus Wiseman_, Sep 30 2019