A060072 -id:A060072 - cp's OEIS frontend

A157406 The integer partitions of n taken as digits in base n+1 and listed in the Hindenburg order.

0, 1, 2, 4, 3, 9, 21, 4, 16, 12, 56, 156, 5, 25, 20, 115, 85, 475, 1555, 6, 36, 30, 204, 24, 162, 1086, 114, 792, 5202, 19608, 7, 49, 42, 329, 35, 273, 2121, 217, 210, 1673, 12873, 1169, 9289, 70217, 299593

Offset: 0

Peter Luschny, Mar 11 2009

The rows are enumerated 0,1,2,... Converting the numbers in the n-th row (n>0) to base n+1 gives all partitions of n in the 'Hindenburg order'. The term 'Hindenburg order' is not standard and refers to the partition generating algorithm of C. F. Hindenburg (1779).

The offset of row n (n>0) is A000070[n+1], the length of row n is A000041[n]. The right hand side of the triangle 0,1,4,21,156,... is A060072.

			[0] <-> [[ ]]
[1] <-> [[1]]
[2,4] <-> [[2],[1,1]]
[3,9,21] <-> [[3],[1,2],[1,1,1]]
[4,16,12,56,156] <-> [[4],[1,3],[2,2],[1,1,2],[1,1,1,1]]

Peter Luschny, Counting with Partitions.

Cf. A157407

a := proc(n) local rev,P,R,i,l,s,k,j;
rev := l -> [seq(l[nops(l)-j+1],j=1..nops(l))];
P := rev(combinat[partition](n)); R := NULL;
for i to nops(P) do l := convert(P[i],base,n+1,10);
s := add(l[k]*10^(k-1),k=1..nops(l));
R := R,s; od; R end: [0,seq(a(i),i=1..7)];

A157407 The integer partitions of n taken as digits in base n+1 and listed in the reflected Hindenburg order.

Original entry on oeis.org

0, 1, 4, 2, 21, 6, 3, 156, 32, 12, 8, 4, 1555, 260, 50, 45, 15, 10, 5, 19608, 2802, 408, 114, 402, 66, 24, 60, 18, 12, 6, 299593, 37450, 4690, 658, 4683, 595, 147, 91, 588, 84, 28, 77, 21, 14, 7

Offset: 0

Peter Luschny, Mar 11 2009

The rows are enumerated 0,1,2,... Converting the numbers in the n-th row (n>0) to base n+1 gives all partitions of n in the 'reflected Hindenburg order'. The term 'reflected Hindenburg order' is not standard and refers to the partition generating algorithm of C. F. Hindenburg (1779).

The offset of row n (n>0) is A000070[n+1], the length of row n is A000041[n]. The left hand side of the triangle 0,1,4,21,156,... is A060072.

			[0] <-> [[ ]]
[1] <-> [[1]]
[4,2] <-> [[1,1],[2]]
[21,6,3] <-> [[1,1,1],[2,1],[3]]
[156,32,12,8,4] <-> [[1,1,1,1],[2,1,1],[2,2],[3,1],[4]]

Peter Luschny, Counting with Partitions.

Cf. A157406

a := proc(n) local rev,P,R,Q,i,l,s,k,j;
rev := l -> [seq(l[nops(l)-j+1],j=1..nops(l))];
P := combinat[partition](n); R := NULL;
for i to nops(P) do Q := rev(P[i]);
l := convert(Q,base,n+1,10);
s := add(l[k]*10^(k-1), k=1..nops(l));
R:= R,s; od; R end: [0,seq(a(i),i=1..7)];

A175151 a(n) = Sum_{i=1..n} ((i+1)^i - 1)/i.

Original entry on oeis.org

1, 5, 26, 182, 1737, 21345, 320938, 5701778, 116812889, 2710555349, 70256770866, 2011763864406, 63066746422417, 2148275748236033, 79009709388692498, 3120334201617871778, 131703367127423550129, 5916556161455825857509, 281857608793034773225930

Offset: 1

Ctibor O. Zizka, Feb 26 2010

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A000169, A023037, A037205, A060072.

[(&+[((j+1)^j -1)/j: j in [1..n]]): n in [1..30]]; // G. C. Greubel, Aug 16 2022

Accumulate[Table[((i+1)^i-1)/i,{i,20}]] (* Harvey P. Dale, Jul 08 2017 *)

[sum(((j+1)^j -1)/j for j in (1..n)) for n in (1..30)] # G. C. Greubel, Aug 16 2022

a(n) = Sum_{i=1..n+1} A060072(i). - R. J. Mathar, Mar 05 2010

a(n) = Sum_{j=1..n} (j+1)^j/j - H(n), where H(n) is the n-th harmonic number. - G. C. Greubel, Aug 16 2022

More terms from R. J. Mathar, Mar 05 2010

A215159 a(n) = floor(n^n / (n+1)).

Original entry on oeis.org

1, 0, 1, 6, 51, 520, 6665, 102942, 1864135, 38742048, 909090909, 23775972550, 685853880635, 21633936185160, 740800455037201, 27368368148803710, 1085102592571150095, 45957792327018709120, 2070863582910344082917, 98920982783015679456198

Offset: 0

Alex Ratushnyak, Aug 04 2012

b(n) = n^n mod (n+1) begins: 0, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A060072 is essentially floor((n+1)^n / n).
Cf. A173499 is  equal to   floor((n-1)^n / n).
Cf. A023037 is essentially floor((n+1)^(n+1) / n).

[Floor(n^n/(n+1)): n in [0..30]]; // G. C. Greubel, Aug 16 2022

Table[If[n==0, 1, Floor[n^n/(n+1)]], {n,0,30}] (* G. C. Greubel, Aug 16 2022 *)

for n in range(55):
    print(n**n // (n+1), end=",")

[(n^n//(n+1)) for n in (0..30)] # G. C. Greubel, Aug 16 2022

A177164 a(n) = (n^r - 1)/r^2, where r = (n^(n-1) - 1)/(n-1).

Original entry on oeis.org

1, 5, 9972894583, 449853889404077636694265177903207995382439448590987815041588427345865911961016023550064137351211162870609

Offset: 2

Alexander Adamchuk, May 04 2010

nonn

The next term has 1204 digits.
r = (n^(n-1) - 1)/(n-1) = A060072(n) is the (n-1)-digit repunit in base n.
r^2 divides n^r - 1 for all bases n > 1.

			a(10) = (10^111111111 - 1)/111111111^2.

Cf. A060072, A127100, A127101, A127102, A127103, A127104, A127105, A127106, A127107, A127092.

Table[(n^((n^(n - 1) - 1)/(n - 1)) - 1)/((n^(n - 1) - 1)/(n - 1))^2, {n, 2, 6}]

a(n) = (n^((n^(n-1) - 1)/(n-1)) - 1)/((n^(n-1) - 1)/(n-1))^2.

a(n) = (n^A060072(n) - 1)/A060072(n)^2.

cp's OEIS Frontend

A157406 The integer partitions of n taken as digits in base n+1 and listed in the Hindenburg order.

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A157407 The integer partitions of n taken as digits in base n+1 and listed in the reflected Hindenburg order.

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A175151 a(n) = Sum_{i=1..n} ((i+1)^i - 1)/i.

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A215159 a(n) = floor(n^n / (n+1)).

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A177164 a(n) = (n^r - 1)/r^2, where r = (n^(n-1) - 1)/(n-1).

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