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a(n) = A083710(n) - A000041(n-1).
15

%I #29 Feb 15 2023 10:47:18

%S 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 1,181,1,196,45,233,17,389,1,387,80,545,1,750,1,839,165,1004,1,1516,

%U 12,1612,234,2040,1,2766,48,3142,388,3720,1,5295,1,5606,663,7038,83,9194,1,10379,1005

%N a(n) = A083710(n) - A000041(n-1).

%C 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 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 Alois P. Heinz, <a href="/A083711/b083711.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{ d|n, d<n} A000041(d-1).

%e From _Gus Wiseman_, Apr 18 2021: (Start)

%e The a(6) = 4 through a(12) = 13 partitions:

%e (6) (7) (8) (9) (10) (11) (12)

%e (3,3) (4,4) (6,3) (5,5) (6,6)

%e (4,2) (6,2) (3,3,3) (8,2) (8,4)

%e (2,2,2) (4,2,2) (4,4,2) (9,3)

%e (2,2,2,2) (6,2,2) (10,2)

%e (4,2,2,2) (4,4,4)

%e (2,2,2,2,2) (6,3,3)

%e (6,4,2)

%e (8,2,2)

%e (3,3,3,3)

%e (4,4,2,2)

%e (6,2,2,2)

%e (4,2,2,2,2)

%e (2,2,2,2,2,2)

%e (End)

%p 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 A. Sellers_, Jun 21 2003

%p # second Maple program:

%p a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):

%p seq(a(n), n=1..69); # _Alois P. Heinz_, Feb 15 2023

%t a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 15 2023 *)

%Y Allowing 1's gives A083710.

%Y The strict case is A098965.

%Y The complement (except also without 1's) is counted by A338470.

%Y The dual version is A339619.

%Y A000005 counts divisors.

%Y A000041 counts partitions.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A018818 counts partitions into divisors (strict: A033630).

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y A339564 counts factorizations with a selected factor.

%Y Cf. A001787, A001792, A015723, A097986, A098743, A130689, A130714, A264401, A339563, A342193.

%K nonn,easy

%O 1,4

%A _N. J. A. Sloane_, Jun 16 2003

%E More terms from _James A. Sellers_, Jun 21 2003