A060072 -id:A060072 - cp's OEIS frontend

A125118 Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

1, 3, 4, 7, 13, 21, 15, 40, 85, 156, 31, 121, 341, 781, 1555, 63, 364, 1365, 3906, 9331, 19608, 127, 1093, 5461, 19531, 55987, 137257, 299593, 255, 3280, 21845, 97656, 335923, 960800, 2396745, 5380840, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111

Offset: 1

Reinhard Zumkeller, Nov 21 2006

			First 4 rows:
1: [1]_2
2: [11]_2 ........ [11]_3
3: [111]_2 ....... [111]_3 ....... [111]_4
4: [1111]_2 ...... [1111]_3 ...... [1111]_4 ...... [1111]_5
_
1: 1
2: 2+1 ........... 3+1
3: (2+1)*2+1 ..... (3+1)*3+1 ..... (4+1)*4+1
4: ((2+1)*2+1)*2+1 ((3+1)*3+1)*3+1 ((4+1)*4+1)*4+1 ((5+1)*5+1)*5+1.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Repunit

This triangle shares some features with triangle A104878.
This triangle is a portion of rectangle A055129.
Each term of A110737 comes from the corresponding row of this triangle.
Diagonals (adjusting offset as necessary): A060072, A023037, A031973, A173468.
Columns (adjusting offset as necessary): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
Cf. A023037, A031973, A125119, A125120 (row sums).

[((k+1)^n -1)/k : k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 15 2022

Table[((k+1)^n -1)/k, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Aug 15 2022 *)

def A125118(n,k): return ((k+1)^n -1)/k
flatten([[A125118(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Aug 15 2022

T(n, k) = Sum_{i=0..n-1} (k+1)^i.
T(n+1, k) = (k+1)*T(n, k) + 1.
Sum_{k=1..n} T(n, k) = A125120(n).
T(2*n-1, n) = A125119(n).
T(n, 1) = A000225(n).
T(n, 2) = A003462(n) for n>1.
T(n, 3) = A002450(n) for n>2.
T(n, 4) = A003463(n) for n>3.
T(n, 5) = A003464(n) for n>4.
T(n, 9) = A002275(n) for n>8.
T(n, n) = A060072(n+1).
T(n, n-1) = A023037(n) for n>1.
T(n, n-2) = A031973(n) for n>2.
T(n, k) = A055129(n, k+1) = A104878(n+k, k+1), 1<=k<=n. - Mathew Englander, Dec 19 2020

A037205 a(n) = (n+1)^n - 1.

Original entry on oeis.org

0, 1, 8, 63, 624, 7775, 117648, 2097151, 43046720, 999999999, 25937424600, 743008370687, 23298085122480, 793714773254143, 29192926025390624, 1152921504606846975, 48661191875666868480, 2185911559738696531967, 104127350297911241532840, 5242879999999999999999999, 278218429446951548637196400, 15519448971100888972574851071

Offset: 0

N. J. A. Sloane

For n >= 1, a(n) = order of Fibonacci group F(n+1,n).
The terms, written in base n+1, are n digits of value n. For example, a(4) = 624 = 4444 in base 5. - Marc Morgenegg, Nov 30 2016
For n >= 1, in a square grid of side n, this is the number of ways to populate the grid with 1 X 1 blocks (with at least one block) so that no block falls under the effect of gravity. - Paolo Xausa, Apr 12 2021
For n > 1, (n-1)^2 | a(n). - David A. Corneth, Dec 15 2022

D. L. Johnson, Presentation of Groups, Cambridge, 1976, p. 182.
Richard M. Thomas, The Fibonacci groups revisited, in Groups - St. Andrews 1989, Vol. 2, 445-454, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.

G. C. Greubel, Table of n, a(n) for n = 0..385
Michael Penn, A divisibility problem., YouTube video, 2021.

A diagonal of A202624.

[(n + 1)^n - 1: n in [0..25]]; // G. C. Greubel, Nov 10 2017

Table[(n + 1)^n - 1, {n, 0, 21}] (* or *)
Table[If[n < 1, Length@ #, FromDigits[#, n + 1]] &@ ConstantArray[n, n], {n, 0, 21}] (* Michael De Vlieger, Nov 30 2016 *)

for(n=0,25, print1((n + 1)^n - 1, ", ")) \\ G. C. Greubel, Nov 10 2017

a(n) = A000169(n+1) - 1 = A060072(n+1)*(n-1) = A060073(n+1)*(n-1)^2.
E.g.f.: 1/(exp(LambertW(-x)) - x) - exp(x). - Ilya Gutkovskiy, Nov 30 2016
E.g.f.: -exp(x) - 1/(x + x/LambertW(-x)). - Vaclav Kotesovec, Dec 05 2016
a(n) = Sum_{k=1..n} binomial(n,k)*n^k [from Paolo Xausa's comment]. - Joerg Arndt, Apr 12 2021

Revised by N. J. A. Sloane, Dec 30 2011

A060073 a(n) = (n^(n-1)-1)/(n-1)^2.

Original entry on oeis.org

1, 2, 7, 39, 311, 3268, 42799, 672605, 12345679, 259374246, 6140565047, 161792257795, 4696537119847, 148943500129544, 5124095576030431, 190082780764323705, 7563707819165039903, 321380710796022350410, 14523213296398891966759, 695546073617378871592991

Offset: 2

Henry Bottomley, Feb 21 2001

nonn

Written in base n, a(n) has n-2 digits and looks like 12345... except that the final digit is n-1 rather than n-2.

Note that 2^m-1 divides a(m+1) = ((m+1)^m-1)/m^2 if and only if m = 2^k-1 with gcd(k,m) = 1. Mersenne numbers M = 2^p-1 such that a(M+1)/(2^M-1) is prime are Mersenne primes 2^3-1 = 7 and 2^7-1 = 127. - Thomas Ordowski, Sep 19 2021

			a(10) = 999999999/81 = 111111111/9 = 12345679.

Harry J. Smith, Table of n, a(n) for n = 2..200
Index entries for sequences related to final digits of numbers

Cf. A000142, A037205, A058128, A059522, A060072, A127837 (numbers p such that a(p+1) is prime).

Table[(n^(n - 1) - 1)/(n - 1)^2, {n, 2, 20}] (* Michael De Vlieger, Oct 28 2021 *)

a(n) = { (n^(n - 1) - 1)/(n - 1)^2 } \\ Harry J. Smith, Jul 01 2009

a(n) = A037205(n-1)/(n-1)^2 = A060072(n)/(n-1) = A058128(n)/n = A059522(n)/A000142(n).

A110737 Row sums in A112668.

Original entry on oeis.org

1, 4, 21, 40, 1555, 3906, 299593, 3280, 87381, 435848050, 67546215517, 61035156, 61054982558011, 328114698808274, 76861433640456465, 21523360, 128583032925805678351, 953674316406, 275941052631578947368421, 1743392200

Offset: 1

Amarnath Murthy, Aug 10 2005

For all n, a(n) is a term in row n of triangle A125118, and furthermore if n is prime then a(n) = A060072(n+1). - Mathew Englander, Dec 20 2020

Michael De Vlieger, Table of n, a(n) for n = 1..388

Cf. A110738, A110739, A125118, A060072.

A110737 := proc(n) local i,a ; if n = 1 then RETURN(1) ; else a := 2 ; while (1-a^n)/(1-a) mod n <> 0 do a := a+1 ; od ; RETURN( (1-a^n)/(1-a) ) ; fi ; end: for n from 1 to 25 do printf("%d, ",A110737(n)) : od : # R. J. Mathar, Mar 13 2007

Block[{a = {1}, k, s}, Do[k = 2; While[Mod[Set[s, Total@ NestList[# k &, 1, i - 1]], i] != 0, k++]; AppendTo[a, s], {i, 2, 20}]; a] (* Michael De Vlieger, Dec 31 2020 *)

More terms from R. J. Mathar, Mar 13 2007

A127837 Numbers k such that ((k+1)^k-1)/k^2 is a prime.

Original entry on oeis.org

2, 3, 5, 17, 4357

Offset: 1

Farideh Firoozbakht and Alexander Adamchuk, Mar 13 2007

All terms are primes. Corresponding primes of the form ((k+1)^k-1)/k^2 are listed in A128466 = 2, 7, 311, 7563707819165039903, ... .
It seems that if p is in the sequence then the first three numbers k such that k^2 divides (p+1)^k-1 are: 1, p & ((p+1)^p-1)/p. 2 is in the sequence and the first three terms of A127103 are : 1, 2 & ((2+1)^2-1)/2; 3 is in the sequence and the first three terms of A127104 are : 1, 3 & ((3+1)^3-1)/3; 5 is in the sequence and the first three terms of A127106 are : 1, 5 & ((5+1)^5-1)/5.
No other terms below 20000. - Max Alekseyev, Apr 25 2007

			4357 is in the sequence because (4358^4357-1)/4357^2 is prime.

Cf. A128466, A037205, A060072, A060073, A058128, A128456, A127103, A127104, A127106, A128398.

A128466 Primes of the form ((k+1)^k - 1)/k^2 = A060073(k+1).

Original entry on oeis.org

2, 7, 311, 7563707819165039903

Offset: 1

Alexander Adamchuk, Mar 09 2007

nonn

Corresponding numbers k are listed in A127837.
Terms are the primes in A060073.
Next term has 15850 = 1 + floor((4357*log(4358) - 2*log(4357))/log(10)) digits and is too large to include. - M. F. Hasler, May 22 2007

Cf. A127837, A037205, A060072, A060073, A058128, A128456

Select[Table[((n+1)^n-1)/n^2,{n,500}],PrimeQ]  (* Harvey P. Dale, Apr 30 2011 *)

A128466(n)=A060073(A127837(n)+1) /* see there. --- or: */ forprime(p=1,10^5,if(ispseudoprime(n=((p+1)^p-1)/p^2),print1(n,", "))); \\ M. F. Hasler, May 22 2007

a(n) = ((A127837(n) + 1)^A127837(n) - 1) / A127837(n)^2.

A068792 a(n) = (n-1)n^(n-2) + Sum_{i=1..n} (n-i)(n^(n-i-1) + n^(n+i-3)).

Original entry on oeis.org

1, 16, 441, 24336, 2418025, 384473664, 89755965649, 28953439105600, 12345678987654321, 6727499948806851600, 4562491230669011577289, 3769449794266138309731600, 3727710895159027432980276121, 4348096581244536814777202995456, 5907679981266292758213173560296225

Offset: 2

Reinhard Zumkeller, Mar 04 2002

nonn

a(n) is a palindrome in base n representation for all n.

			a(8) = 89755965649 = (1234567654321)OCT;
a(10) = 12345678987654321 = A057139(9);
a(16) = 5907679981266292758213173560296225 = (123456789ABC...987654321)HEX.

G. C. Greubel, Table of n, a(n) for n = 2..215

Cf. A023811, A060072, A062813, A068793.

[((n^(n-1) -1)/(n-1))^2: n in [2..30]]; // G. C. Greubel, Aug 16 2022

Table[((n^(n-1) -1)/(n-1))^2, {n,2,30}] (* G. C. Greubel, Aug 16 2022 *)

def A068792(n): return ((n**(n-1)-1)//(n-1))**2 # Chai Wah Wu, Mar 18 2024

[((n^(n-1) -1)/(n-1))^2 for n in (2..30)] # G. C. Greubel, Aug 16 2022

a(n) = ( (n^(n-1) - 1)/(n-1) )^2.
a(n) = ((A023811(n) - n + 1)/n)*n^(n-1) + A062813(n)/n.
a(n) = A060072(n)^2.

More terms from G. C. Greubel, Aug 16 2022

A125598 a(n) = ((n+1)^(n-1) - 1)/n.

Original entry on oeis.org

0, 1, 5, 31, 259, 2801, 37449, 597871, 11111111, 235794769, 5628851293, 149346699503, 4361070182715, 139013933454241, 4803839602528529, 178901440719363487, 7143501829211426575, 304465936543600121441

Offset: 1

Alexander Adamchuk, Nov 26 2006

nonn

Odd prime p divides a(p-2).
a(n) is prime for n = {3,4,6,74, ...}; prime terms are {5, 31, 2801, ...}.
a(n) is the (n-1)-th generalized repunit in base (n+1). For example, a(5) = 259 which is 1111 in base 6. - Mathew Englander, Oct 20 2020

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

Cf. A000272 (n^(n-2)), A125599.

Cf. other sequences of generalized repunits, such as A125118, A053696, A055129, A060072, A031973, A173468, A023037, A119598, A085104, and A162861.

[((n+1)^(n-1) -1)/n: n in [1..25]]; // G. C. Greubel, Aug 15 2022

Mathematica
```
Table[((n+1)^(n-1)-1)/n, {n,25}]
```

[gaussian_binomial(n,1,n+2) for n in range(0,18)] # Zerinvary Lajos, May 31 2009

a(n) = ((n+1)^(n-1) - 1)/n.
a(n) = (A000272(n+1) - 1)/n.
a(2k-1)/(2k+1) = A125599(k) for k>0.
From Mathew Englander, Dec 17 2020: (Start)
a(n) = (A060072(n+1) - A083069(n-1))/2.
For n > 1, a(n) = Sum_{k=0..n-2} (n+1)^k.
For n > 1, a(n) = Sum_{j=0..n-2} n^j*C(n-1,j+1). (End)

A077385 Triangle read by rows in which n-th row contains 2n-1 terms starting from n^0 to n^(n-1) in increasing order and then in decreasing order to n^0.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 9, 3, 1, 1, 4, 16, 64, 16, 4, 1, 1, 5, 25, 125, 625, 125, 25, 5, 1, 1, 6, 36, 216, 1296, 7776, 1296, 216, 36, 6, 1, 1, 7, 49, 343, 2401, 16807, 117649, 16807, 2401, 343, 49, 7, 1, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 262144, 32768, 4096, 512, 64, 8, 1

Offset: 1

Amarnath Murthy, Nov 06 2002

			Irregular triangle begins as:
  1;
  1, 2,  1;
  1, 3,  9,   3,    1;
  1, 4, 16,  64,   16,     4,      1;
  1, 5, 25, 125,  625,   125,     25,     5,    1;
  1, 6, 36, 216, 1296,  7776,   1296,   216,   36,   6,  1;
  1, 7, 49, 343, 2401, 16807, 117649, 16807, 2401, 343, 49, 7, 1;

G. C. Greubel, Rows n = 1..40 of the irregular triangle, flattened

Cf. A000169, A000272, A023037, A060072, A077386.

A077385:= func< n,k | k lt n select n^k else n^(2*n-k-2) >;
[A077385(n,k): k in [0..2*n-2], n in [1..12]]; // G. C. Greubel, Sep 21 2022

A077385 := proc(n,k) if k < n then n^k ; else n^(2*n-k-2) ; fi ; end: for n from 1 to 10 do for k from 0 to 2*n-2 do printf("%d, ",A077385(n,k)) ; od : od : # R. J. Mathar, Jul 03 2007

Table[Join[n^Range[0,n-1],n^Range[n-2,0,-1]],{n,8}]//Flatten (* Harvey P. Dale, Oct 13 2017 *)

def A077385(n,k): return n^k if (kA077385(n,k) for k in (0..2*n-2)] for n in (1..12)]) # G. C. Greubel, Sep 21 2022

T(n, k) = n^k for k < n, otherwise n^(2*n-k-2), for n >= 1, 0 <= k <= 2*n-2.
From G. C. Greubel, Sep 21 2022: (Start)
T(n, 0) = T(n, 2*n-2) = 1.
T(n, n-1) = A000169(n).
T(n, n) = A000272(n).
T(n, 2*n-2-k) = T(n, k).
Sum_{k=0..n-1} T(n, k) = A023037(n).
Sum_{k=0..n-2} T(n, k) = A060072(n).
Sum_{k=0..2*n-2} T(n, k) = A077386(n) = 2*A060072(n) + A000169(n), n > 1. (End)

More terms from R. J. Mathar, Jul 03 2007

A065583 Sum of numbers which in base n have (n-1) distinct nonzero digits.

Original entry on oeis.org

0, 1, 12, 252, 9360, 559800, 49412160, 6039794880, 976299609600, 201599999798400, 51766949513664000, 16177372653293913600, 6044902527410562816000, 2661334524326601925401600, 1363387181797265578297344000, 804077813274862776803112960000, 540880443323184957954046525440000

Offset: 1

Henry Bottomley, Nov 28 2001

			a(4) = 252 since we need to sum the base 4 numbers 123, 132, 213, 231, 312 and 321, i.e. the decimal sum 27+30+39+45+54+57 = 252.

G. C. Greubel, Table of n, a(n) for n = 1..230

Cf. A001710, A060072.

[n eq 1 select 0 else Factorial(n)*(n^(n-1)-1)/(2*(n-1)): n in [1..30]]; // G. C. Greubel, Aug 16 2022

Table[If[n==1, 0, n!*(n^(n-1) -1)/(2*(n-1))], {n,30}] (* G. C. Greubel, Aug 16 2022 *)

[0]+[factorial(n)*(n^(n-1)-1)/(2*(n-1)) for n in (2..30)] # G. C. Greubel, Aug 16 2022

a(n) = n!*(n^(n-1)-1)/(2*(n-1)) = A001710(n)*A060072(n).

More terms from Benoit Cloitre, Jan 31 2002

More terms from G. C. Greubel, Aug 16 2022

cp's OEIS Frontend

A125118 Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

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Magma

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A037205 a(n) = (n+1)^n - 1.

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A060073 a(n) = (n^(n-1)-1)/(n-1)^2.

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A110737 Row sums in A112668.

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Maple

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A127837 Numbers k such that ((k+1)^k-1)/k^2 is a prime.

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A128466 Primes of the form ((k+1)^k - 1)/k^2 = A060073(k+1).

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Mathematica

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A068792 a(n) = (n-1)*n^(n-2) + Sum_{i=1..n} (n-i)*(n^(n-i-1) + n^(n+i-3)).

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Magma

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SageMath

A068792 a(n) = (n-1)n^(n-2) + Sum_{i=1..n} (n-i)(n^(n-i-1) + n^(n+i-3)).