A371550
Expansion of e.g.f. Product_{k>=1} (1 + mu(k)^2*x^k/k!).
Original entry on oeis.org
1, 1, 1, 4, 4, 11, 67, 29, 260, 876, 3841, 34134, 69774, 152231, 774243, 4182754, 30376720, 409813561, 1056300594, 3175397668, 3655126844, 91668397027, 499871922705, 5219638765816, 120716816247428, 17518596045460, 193032439198301, 3860666483055372, 22675538336311998
Offset: 0
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nmax = 28; CoefficientList[Series[Product[(1 + MoebiusMu[k]^2 x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A378843
Number of compositions (ordered partitions) of n into distinct squarefree divisors of n.
Original entry on oeis.org
1, 1, 1, 1, 0, 1, 7, 1, 0, 0, 1, 1, 24, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 151, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 31, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 864, 1, 1, 0, 0, 1, 127, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 7, 1, 0
Offset: 0
a(6) = 7 because we have [6], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].
a(12) = 24 because we have [6, 3, 2, 1] and 4! = 24 permutations.
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ptns:= proc(S,n) option remember;
# subsets of S with sum n
local m,s;
if convert(S,`+`) < n then return {} fi;
if n = 0 then return {{}} fi;
s:= max(S);
if s > n then return procname(select(`<=`,S,n),n) fi;
map(t -> t union {s}, procname(S minus {s},n-s)) union procname(S minus {s}, n)
end proc:
sfd:= proc(n) map(convert,combinat:-powerset(numtheory:-factorset(n)),`*`) end proc:
f:= proc(n) local t;
add((nops(t))!, t = ptns(sfd(n),n))
end proc:
map(f, [$0..100]); # Robert Israel, Dec 15 2024
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a[n_] := Module[{d = Select[Divisors[n], SquareFreeQ]}, Total[(Length /@ Select[Subsets[d], Total[#] == n &])!]]; a[0] = 1; Array[a, 100, 0] (* Amiram Eldar, Dec 10 2024 *)
A281667
Expansion of Sum_{i>=1} mu(i)^2*x^i/(1 + x^i) * Product_{j>=1} (1 + mu(j)^2*x^j), where mu() is the Moebius function (A008683).
Original entry on oeis.org
1, 1, 3, 2, 3, 6, 5, 9, 10, 12, 15, 16, 20, 24, 27, 38, 41, 48, 56, 62, 78, 88, 101, 120, 131, 149, 174, 189, 221, 243, 278, 318, 349, 394, 444, 491, 556, 622, 693, 773, 849, 953, 1048, 1158, 1292, 1422, 1568, 1735, 1901, 2101, 2307, 2534, 2795, 3060, 3357, 3681, 4024, 4404, 4809, 5245, 5734, 6242, 6805, 7418
Offset: 1
a(8) = 9 because we have [7, 1], [6, 2], [5, 3], [5, 2, 1] and 2 + 2 + 2 + 3 = 9.
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nmax = 64; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i/(1 + x^i), {i, 1, nmax}] Product[1 + MoebiusMu[j]^2 x^j, {j, 1, nmax}], {x, 0, nmax}], x]]
A305080
Number of connected strict integer partitions of n with pairwise indivisible and squarefree parts.
Original entry on oeis.org
1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 2, 2, 3, 2, 2, 4, 2, 3, 4, 4, 3, 4, 3, 4, 5, 6, 4, 6, 5, 7, 6, 5, 6, 8, 6, 6, 6, 10, 11, 11, 9, 11, 9, 13
Offset: 1
The a(52) = 6 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,10,6): {{2,4},{2,3},{1,3},{1,2}}
(22,14,10,6): {{1,5},{1,4},{1,3},{1,2}}
(30,22): {{1,2,3},{1,5}}
(38,14): {{1,8},{1,4}}
(42,10): {{1,2,4},{1,3}}
(46,6): {{1,9},{1,2}}
The a(60) = 8 strict partitions together with their corresponding multiset multisystems (which are clutters):
(21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
(33,21,6): {{2,5},{2,4},{1,2}}
(35,15,10): {{3,4},{2,3},{1,3}}
(39,15,6): {{2,6},{2,3},{1,2}}
(34,26): {{1,7},{1,6}}
(38,22): {{1,8},{1,5}}
(39,21): {{2,6},{2,4}}
(46,14): {{1,9},{1,4}}
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Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@SquareFreeQ/@#,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,50}]
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