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A083711 a(n) = A083710(n) - A000041(n-1).

%I A083711 #30 Jul 02 2025 16:02:01
%S A083711 1,1,1,2,1,4,1,5,3,7,1,14,1,13,8,20,1,33,1,40,14,44,1,85,6,79,25,117,
%T A083711 1,181,1,196,45,233,17,389,1,387,80,545,1,750,1,839,165,1004,1,1516,
%U A083711 12,1612,234,2040,1,2766,48,3142,388,3720,1,5295,1,5606,663,7038,83,9194,1,10379,1005
%N A083711 a(n) = A083710(n) - A000041(n-1).
%C A083711 Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - _Gus Wiseman_, Apr 18 2021
%D A083711 L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.
%H A083711 Alois P. Heinz, <a href="/A083711/b083711.txt">Table of n, a(n) for n = 1..10000</a>
%F A083711 a(n) = Sum_{ d|n, d<n} A000041(d-1).
%e A083711 From _Gus Wiseman_, Apr 18 2021: (Start)
%e A083711 The a(6) = 4 through a(12) = 13 partitions:
%e A083711   (6)      (7)  (8)        (9)      (10)         (11)  (12)
%e A083711   (3,3)         (4,4)      (6,3)    (5,5)              (6,6)
%e A083711   (4,2)         (6,2)      (3,3,3)  (8,2)              (8,4)
%e A083711   (2,2,2)       (4,2,2)             (4,4,2)            (9,3)
%e A083711                 (2,2,2,2)           (6,2,2)            (10,2)
%e A083711                                     (4,2,2,2)          (4,4,4)
%e A083711                                     (2,2,2,2,2)        (6,3,3)
%e A083711                                                        (6,4,2)
%e A083711                                                        (8,2,2)
%e A083711                                                        (3,3,3,3)
%e A083711                                                        (4,4,2,2)
%e A083711                                                        (6,2,2,2)
%e A083711                                                        (4,2,2,2,2)
%e A083711                                                        (2,2,2,2,2,2)
%e A083711 (End)
%p A083711 with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d,`,a(j)) od: # _James Sellers_, Jun 21 2003
%p A083711 # second Maple program:
%p A083711 a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
%p A083711 seq(a(n), n=1..69);  # _Alois P. Heinz_, Feb 15 2023
%t A083711 a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
%t A083711 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 15 2023 *)
%Y A083711 Allowing 1's gives A083710.
%Y A083711 The strict case is A098965.
%Y A083711 The complement (except also without 1's) is counted by A338470.
%Y A083711 The dual version is A339619.
%Y A083711 A000005 counts divisors.
%Y A083711 A000041 counts partitions.
%Y A083711 A000070 counts partitions with a selected part.
%Y A083711 A006128 counts partitions with a selected position.
%Y A083711 A018818 counts partitions into divisors (strict: A033630).
%Y A083711 A167865 counts strict chains of divisors > 1 summing to n.
%Y A083711 A339564 counts factorizations with a selected factor.
%Y A083711 Cf. A001787, A001792, A015723, A097986, A098743, A130689, A130714, A264401, A339563, A342193.
%K A083711 nonn,easy
%O A083711 1,4
%A A083711 _N. J. A. Sloane_, Jun 16 2003
%E A083711 More terms from _James Sellers_, Jun 21 2003