A065033 -id:A065033 - cp's OEIS frontend

A258289 Number of partitions of 1, 2, 3, or more copies of n into distinct parts.

1, 1, 1, 3, 3, 7, 9, 17, 21, 43, 57, 109, 157, 301, 447, 895, 1307, 2663, 4207, 8463, 13283, 28489, 45151, 95485, 157767, 336711, 561603, 1236963, 2061173, 4567227, 7946575, 17516101, 30324977, 69519697, 121465499, 276609723, 496333307, 1137900605

Offset: 0

Alois P. Heinz, May 25 2015

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [2].
a(3) = 3: [3], [2,1], [3;2,1].
a(4) = 3: [4], [3,1], [4;3,1].
a(5) = 7: [5], [4,1], [3,2], [5;4,1], [5;3,2], [4,1;3,2], [5;4,1;3,2].
a(7) = 17: [7], [6,1], [5,2], [4,3], [4,2,1], [7;6,1], [7;5,2], [7;4,3], [7;4,2,1], [6,1;5,2], [6,1;4,3], [5,2;4,3], [7;6,1;5,2], [7;6,1;4,3], [7;5,2;4,3], [6,1;5,2;4,3], [7;6,1;5,2;4,3].

Cf. A000009, A065033, A258280.

b:= proc() option remember; local m; m:= args[nargs];
     `if`(nargs=1, 1, `if`(args[1]=0, b(args[t] $t=2..nargs),
     `if`(m=0 or add(args[i], i=1..nargs-1)> m*(m+1)/2, 0,
      b(args[t] $t=1..nargs-1, m-1)+add(`if`(args[j]-m<0, 0,
      b(sort([seq(args[i]-`if`(i=j, m, 0), i=1..nargs-1)])[]
      , m-1)), j=1..nargs-1))))
    end:
a:= n-> add(b(n$k+1)/k!, k=1..max(1, ceil(n/2))):
seq(a(n), n=0..20);

disParts[n_] := disParts[n] = Select[IntegerPartitions[n], Length[#] == Length[Union[#]]&];
T[n_, k_] := Select[Subsets[disParts[n], {k}], Length[Flatten[#]] == Length[Union[Flatten[#]]]&] // Length;
a[n_] := a[n] = If[n == 0, 1, Sum[T[n, k], {k, 1, Quotient[n+1, 2]}]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, May 01 2022 *)

a(n) = Sum_{k=1..A065033(n)} A258280(n,k).

a(n) = Sum_{k=1..max(1,ceiling(n/2))} 1/k! * [Product_{i=1..k} x_i^n] Product_{j>0} (1+Sum_{i=1..k} x_i^j).

A363377 Largest positive integer having n holes that can be made using the fewest possible digits.

Original entry on oeis.org

7, 9, 8, 98, 88, 988, 888, 9888, 8888, 98888, 88888, 988888, 888888, 9888888, 8888888, 98888888, 88888888, 988888888, 888888888, 9888888888, 8888888888, 98888888888, 88888888888, 988888888888, 888888888888, 9888888888888, 8888888888888, 98888888888888, 88888888888888, 988888888888888

Offset: 0

Julia Zimmerman, May 29 2023

Each decimal digit has 0, 1 or 2 holes so that n holes requires A065033(n) digits.

			For n=0, the largest integer with no holes in it that is as short as possible is 7 (9 is larger, but has 1 hole; 11 is larger and has no holes, but is longer at length 2 > length 1).
For n=1, the largest integer with 1 hole that is as short as possible is 9 (following the same kind of reasoning as with n=0).

Index entries for sequences related to holes in digits.
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A002281 and A002282 (number of holes), A065033 (digits required).
Cf. A249572 and A250256 (smallest number).
Cf. A337099 (largest 7-segment).

CoefficientList[Series[(7 + 2 x - 71 x^2 + 70 x^3)/((1 - x) (1 - 10 x^2)), {x, 0, 30}], x] (* Michael De Vlieger, Jul 05 2023 *)

A363377=lambda n: (8+n%2*81)*10**(n>>1)//9 if n else 7
print([A363377(n) for n in range(30)]) # Natalia L. Skirrow, Jun 26 2023

From Natalia L. Skirrow, Jun 26 2023: (Start)
a(n) = (89*(10^((n-1)/2))-8)/9 for odd n; a(n) = 8*(10^(n/2)-1)/9 for even n >= 2.
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3), for n >= 4.
G.f.: (7+2*x-71*x^2+70*x^3)/((1-x)*(1-10*x^2)).
E.g.f.: (80*cosh(sqrt(10)*x) + 89*sqrt(10)*sinh(sqrt(10)*x) - 80*e^x)/90 + 7. (End)

A077219 Floor(geometric mean of the reduced residue system modulo n).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 19, 20, 19, 20, 20, 20, 21, 22, 22, 23, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 27, 28, 28

Offset: 1

Joseph L. Pe, Nov 30 2002

nonn

1. The reduced residue system modulo n is the set of integers k between 1 and n which are coprime to n. The geometric mean of the positive integers a_1,...,a_n is the n-th root of a_1*...*a_n. 2. The arithmetic mean of the reduced residue system modulo n is A065033.

gm[l_] := Module[{k, p}, k = Length[l]; p = Product[l[[i]], {i, 1, k}]; p^(1/k)]; rp[n_] := Module[{a, i}, a = {1}; For[i = 2, i < n, i++, If[GCD[i, n] == 1, a = Append[a, i]]]; a];

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A258289 Number of partitions of 1, 2, 3, or more copies of n into distinct parts.

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A363377 Largest positive integer having n holes that can be made using the fewest possible digits.

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A077219 Floor(geometric mean of the reduced residue system modulo n).

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