A037205 -id:A037205 - cp's OEIS frontend

A060071 Duplicate of A037205.

0, 1, 8, 63, 624, 7775, 117648, 2097151, 43046720, 999999999, 25937424600

Offset: 1

Henry Bottomley, Feb 21 2001

dead

A060072 a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.

0, 1, 4, 21, 156, 1555, 19608, 299593, 5380840, 111111111, 2593742460, 67546215517, 1941507093540, 61054982558011, 2085209001813616, 76861433640456465, 3041324492229179280, 128583032925805678351, 5784852794328402307380, 275941052631578947368421

Offset: 1

Henry Bottomley, Feb 21 2001

nonn

(n-1)-digit repunits in base n written in decimal.

			a(10)=111111111; i.e., just nine 1's (converted from base 10 to decimal).

Harry J. Smith, Table of n, a(n) for n=1..200

Cf. A000169, A037205, A055869, A060073, A125118, A228275.

Cf. other sequences of generalized repunits, such as A053696, A055129, A031973, A125598, A173468, A023037, A119598, A085104, and A162861.

[0] cat [ (n^(n-1) -1)/(n-1) : n in [2..25]]; // G. C. Greubel, Aug 15 2022

Join[{0},Array[(#^(#-1)-1)/(#-1)&,20,2]] (* Harvey P. Dale, Jun 04 2013 *)

a(n) = if (n==1, 0, (n^(n - 1) - 1)/(n - 1)); \\ Harry J. Smith, Jul 01 2009

[0]+[(n^(n-1) -1)/(n-1) for n in (2..25)] # G. C. Greubel, Aug 15 2022

a(n+1) = Sum_{k=1..n} n^(k-1)*C(n, k). - Olivier Gérard, Jun 26 2001 [Corrected by Mathew Englander, Dec 15 2020]
a(n) = Sum_{j=2..n} n^(n-j). - Zerinvary Lajos, Sep 11 2006
a(n+1) = A125118(n,n). - Reinhard Zumkeller, Nov 21 2006
a(n) = Integral_{x=1/n..1} 1/x^n dx. - Francesco Daddi, Aug 01 2011
a(n) = A037205(n-1)/(n-1) = A060073(n)*(n-1) = A023037(n) - A000169(n).
a(n) = [x^n] x^2/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017
a(n) = 1 + A228275(n, n-2) for n >= 2. - Mathew Englander, Dec 14 2020

Name edited by Michel Marcus, Dec 14 2020

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).

A061190 a(n) = n^n - n.

Original entry on oeis.org

1, 0, 2, 24, 252, 3120, 46650, 823536, 16777208, 387420480, 9999999990, 285311670600, 8916100448244, 302875106592240, 11112006825558002, 437893890380859360, 18446744073709551600, 827240261886336764160, 39346408075296537575406, 1978419655660313589123960

Offset: 0

Lawrence H. Shirley (LShirley(AT)towson.edu), May 30 2001

Number of endofunctions on [n] such that no element has a preimage of cardinality n. - Alois P. Heinz, Jul 21 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Main diagonal of A245405.

[n^n - n : n in [0..20]]; // Vincenzo Librandi, Sep 01 2011

Table[n^n-n, {n,25}] (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)

E.g.f.: 1/(1 + LambertW(-x)) - x*exp(x), where LambertW() is the Lambert W-function. - Ilya Gutkovskiy, Feb 08 2017

a(n) = n*A037205(n-1). - R. J. Mathar, Jul 02 2017

a(0)=1 inserted by Alois P. Heinz, Jul 21 2014

A085606 a(n) = (n-1)^n - 1.

Original entry on oeis.org

0, -1, 0, 7, 80, 1023, 15624, 279935, 5764800, 134217727, 3486784400, 99999999999, 3138428376720, 106993205379071, 3937376385699288, 155568095557812223, 6568408355712890624, 295147905179352825855, 14063084452067724991008, 708235345355337676357631

Offset: 0

Lekraj Beedassy, Jul 07 2003

sign

Sequence relates to the "monkey and coconut problem"(A014293) giving the number of coconuts received by each of the n sailors from the ultimate equitable distribution the next day.
From Alexander Adamchuk, Nov 13 2006: (Start)
4n^2 divides a(2n).
Odd prime p divides a(p-1).
8p^2 divides a(2p) for an odd prime p.
32p^4 divides a(2p^2) for an odd prime p.
64p^8 divides a(2p^4) for an odd prime p.
p^3 divides a(p^3+2) for prime p.
p divides a((p-1)/2) for prime p in A157437.
p^2 divides a((p-1)/2) for prime p = {5,127,607}. (End)

Cf. A014293, A007778, A065440, A037205, A157437.

Table[(n-1)^n-1,{n,0,30}] (* Alexander Adamchuk, Nov 13 2006 *)

a(n) = A065440(n) - 1.

More terms from Ray Chandler, Nov 10 2003

A202624 Array read by antidiagonals: T(n,k) = order of Fibonacci group F(n,k), writing 0 if the group is infinite, for n >= 2, k >= 1.

Original entry on oeis.org

1, 2, 1, 3, 8, 8, 4, 3, 2, 5, 5, 24, 63, 0, 11, 6, 5, 0, 3, 22, 0, 7, 48, 5, 624, 0, 1512, 29, 8, 7, 342, 125, 4, 0, 0, 0, 9, 80, 0, 0, 7775, 0, 0, 0, 0, 10, 9, 8, 7

Offset: 2

N. J. A. Sloane, Dec 29 2011

The Fibonacci group F(r,n) has presentation , where there are n relations, obtained from the first relation by applying the permutation (1,2,,n) to the subscripts and reducing subscripts mod n. Then T(n,k) = |F(n,k)|.

T(7,5) was not known in 1998 (Chalk).

			The array begins:
k =  1  2   3    4    5     6     7     8      9    10 ...
----------------------------------------------------------
n=1: 0  0   0    0    0     0     0     0      0     0 ...
n=2: 1  1   8    5   11     0    29     0      0     0 ...
n=3: 2  8   2    0   22  1512     0     0      0     0 ...
n=4: 3  3  63    3    0     0     0     0      ?     0 ...
n=5: 4 24   0  624    4     0     0     0      0     0 ...
n=6: 5  5   5  125 7775     5     0     0      0     0 ...
n=7: 6 48 342    0    ? 7^6-1     6     0      0     0 ...
n=8: 7  7   0    7    ?     0 8^7-1     7      0     0 ...
n=9: 8 80   8 6560    0     0     0 9^8-1      8     0 ...
n=10 9  9 999 4905    9     ?     ?     0 10^9-1     9 ...
...
For example, T(2,5) = 11, since the presentation <a,b,c,d,e | ab=c, bc=d, cd=e, de=a, ea=b> defines the cyclic group of order 11. This example is due to John Conway.
This table is based on those in Johnson (1976) and Thomas (1989), supplemented by values from Chalk (1998). We have ignored the n=1 row when reading the table by antidiagonals.

Campbell, Colin M.; and Gill, David M. On the infiniteness of the Fibonacci group F(5,7). Algebra Colloq. 3 (1996), no. 3, 283-284.
D. L. Johnson, Presentation of Groups, Cambridge, 1976, see table p. 182.
Mednykh, Alexander; and Vesnin, Andrei; On the Fibonacci groups, the Turk's head links and hyperbolic 3-manifolds, in Groups-Korea '94 (Pusan), 231-239, de Gruyter, Berlin, 1995.
Nikolova, Daniela B., The Fibonacci groups - four years later, in Semigroups (Kunming, 1995), 251-255, Springer, Singapore, 1998.
Nikolova, D. B.; and Robertson, E. F., One more infinite Fibonacci group. C. R. Acad. Bulgare Sci. 46 (1993), no. 3, 13-15.
Thomas, Richard M., 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.

Brunner, A. M., The determination of Fibonacci groups, Bull. Austral. Math. Soc. 11 (1974), 11-14.
A. M. Brunner, On groups of Fibonacci type, Proc. Edinburgh Math. Soc. (2) 20 (1976/77), no. 3, 211-213.
C. M. Campbell and P. P. Campbell, Search techniques and epimorphisms between certain groups and Fibonacci groups, Irish Math. Soc. Bull. No. 56 (2005), 21-28.
Chalk, Christopher P., Fibonacci groups with aspherical presentations, Comm. Algebra 26 (1998), no. 5, 1511-1546.
C. P. Chalk and D. L. Johnson, The Fibonacci groups II, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 12, 79-86.
J. H. Conway et al., Advanced problem 5327, Amer. Math. Monthly, 72 (1965), 915; 74 (1967), 91-93.
Helling, H.; Kim, A. C.; and Mennicke, J. L.; A geometric study of Fibonacci groups, J. Lie Theory 8 (1998), no. 1, 1-23.
Derek F. Holt, An alternative proof that the Fibonacci group F(2,9) is infinite, Experiment. Math. 4 (1995), no. 2, 97-100.
David J. Seal, The orders of the Fibonacci groups, Proc. Roy. Soc. Edinburgh, Sect. A 92 (1982), no. 3-4, 181-192.
A. Szczepanski, The Euclidean representations of the Fibonacci groups, Quart. J. Math. 52 (2001), 385-389.

Cf. A037205 (a diagonal), A065530, A202625, A202626, A202627 (columns).

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.

A246445 Numbers of the form (x^y - x)/y for positive x,y.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 10, 15, 18, 20, 21, 28, 36, 40, 45, 48, 55, 63, 66, 70, 78, 91, 105, 112, 120, 121, 136, 153, 155, 168, 171, 186, 190, 204, 210, 231, 240, 253, 276, 300, 312, 325, 330, 351, 378, 406, 435, 440, 465, 496, 528, 561, 572, 595, 624, 630, 666, 682, 703

Offset: 1

Juri-Stepan Gerasimov, Sep 07 2014

nonn

			8 is in this sequence because (3^3 - 3)/3 = 8.

Juri-Stepan Gerasimov, Table of n, a(n) for n = 1..59

Subsequences of a(n): A000217, A007290, A037205, A208536, A208537.

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

cp's OEIS Frontend

A060071 Duplicate of A037205.

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A060072 a(n) = (n^(n-1) - 1)/(n-1) for n>1, a(1) = 0.

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

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A061190 a(n) = n^n - n.

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A085606 a(n) = (n-1)^n - 1.

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A202624 Array read by antidiagonals: T(n,k) = order of Fibonacci group F(n,k), writing 0 if the group is infinite, for n >= 2, k >= 1.

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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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