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A092976 Consider all partitions of n into parts all of which are divisors of n; a(n) = number of distinct values taken by the product of the parts.

%I A092976 #32 Jul 04 2025 21:49:15
%S A092976 1,1,2,2,3,2,7,2,5,4,10,2,19,2,13,13,9,2,37,2,29,17,19,2,61,6,22,10,
%T A092976 39,2,247,2,17,25,28,25,127,2,31,29,97,2,450,2,59,82,37,2,217,8,146,
%U A092976 37,69,2,271,37,133,41,46,2,1558,2,49,112,33,43,1038,2,89,49,1105,2,469,2,58,211,99,49,1423,2,353
%N A092976 Consider all partitions of n into parts all of which are divisors of n; a(n) = number of distinct values taken by the product of the parts.
%C A092976 a(n) > tau(n) + A(n) + R(n), where tau(n) = number of divisors of n, A(n) = product of powers of nontrivial divisors whose sum with multiplicity is < n and R(n) = numbers of the form r^(k) > n where r is a divisor of n and k <= n/r.
%H A092976 Alois P. Heinz, <a href="/A092976/b092976.txt">Table of n, a(n) for n = 0..10000</a>
%F A092976 a(n) = 2 <=> n in { A000040 }. - _Alois P. Heinz_, Oct 28 2021
%e A092976 a(10) = 10, the numbers arising are 1,2,4,5,8,10,16,20,25 and 32; e.g. 25 = 5*5, 8 = 2*2*2*1*1*1*1, 32 = 2*2*2*2*2, etc.
%p A092976 with(numtheory):
%p A092976 a:= proc(n) local b, l, s;
%p A092976       l:= sort([divisors(n)[]]);
%p A092976       b:= proc(n,i,p)
%p A092976             if n<0 then
%p A092976           elif n=0 then s:= s union {p}
%p A092976           elif i=0 then
%p A092976           else b(n-l[i], i, p*l[i]); b(n, i-1, p)
%p A092976             fi
%p A092976           end;
%p A092976       s:= {};
%p A092976       b(n, nops(l), 1);
%p A092976       nops(s)
%p A092976     end:
%p A092976 seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 21 2011
%t A092976 a[n_] := Module[{ b, l, s}, l = Divisors[n]; b[m_, i_, p_] := Which[m<0, , m == 0, s = Union[s, {p}], i == 0, , True, b[m - l[[i]], i, p*l[[i]]]; b[m, i-1, p]]; s = {}; b[n, Length[l], 1]; Length[s]]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Nov 11 2015, after _Alois P. Heinz_ *)
%Y A092976 Cf. A000040, A092975.
%K A092976 nonn
%O A092976 0,3
%A A092976 _Amarnath Murthy_, Mar 27 2004
%E A092976 More terms from _Vladeta Jovovic_, Aug 06 2005
%E A092976 More terms from _Alois P. Heinz_, Feb 21 2011
%E A092976 a(0)=1 prepended by _Alois P. Heinz_, Oct 28 2021