A023037 -id:A023037 - cp's OEIS frontend

A088790 Numbers k such that (k^k-1)/(k-1) is prime.

2, 3, 19, 31, 7547

Offset: 1

T. D. Noe, Oct 16 2003

Note that (k^k-1)/(k-1) is prime only if k is prime, in which case it equals cyclotomic(k,k), the k-th cyclotomic polynomial evaluated at x=k. This sequence is a subsequence of A070519. The number cyclotomic(7547,7547) is a probable prime found by H. Lifchitz. Are there only a finite number of these primes?
From T. D. Noe, Dec 16 2008: (Start)
The standard heuristic implies that there are an infinite number of these primes and that the next k should be between 10^10 and 10^11.
Let N = (7547^7547-1)/(7547-1) = A023037(7547). If N is prime, then the period of the Bell numbers modulo 7547 is N. See A054767. (End)

R. K. Guy, Unsolved Problems in Theory of Numbers, 1994, A3.

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A070519 (cyclotomic(n, n) is prime).

Cf. A056826 ((n^n+1)/(n+1) is prime).

Do[p=Prime[n]; If[PrimeQ[(p^p-1)/(p-1)], Print[p]], {n, 100}]

is(n)=ispseudoprime((n^n-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

A031972 a(n) = Sum_{k=1..n} n^k.

Original entry on oeis.org

0, 1, 6, 39, 340, 3905, 55986, 960799, 19173960, 435848049, 11111111110, 313842837671, 9726655034460, 328114698808273, 11966776581370170, 469172025408063615, 19676527011956855056, 878942778254232811937, 41660902667961039785742, 2088331858752553232964199

Offset: 0

N. J. A. Sloane, Dec 11 1999

Sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^n: a(n) = Sum_{k=1..n} k*A228273(n,k). a(2) = 6 = 2+1+1+2: [1,1], [1,2], [2,1], [2,2]. - Alois P. Heinz, Aug 19 2013
a(n) is the expected wait time to see the contiguous subword 11...1 (n copies of 1) over all infinite sequences on alphabet {1,2,...,n}. - Geoffrey Critzer, May 19 2014
a(n) is the number of sequences of k elements from {1,2,...,n}, where 1<=k<=n. For example, a(2) = 6, counting the sequences, [1], [2], [1,1], [1,2], [2,1], [2,2]. Equivalently, a(n) is the number of bar graphs having a height and width of at most n. - Emeric Deutsch, Jan 24 2017.
In base n, a(n) has n+1 digits: n 1's followed by a 0. - Mathew Englander, Oct 20 2020

Alois P. Heinz, Table of n, a(n) for n = 0..386
A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.

Main diagonal of A228275.

Cf. A031973, A228273, A023037, A226238.

a031972 n = sum $ take n $ iterate (* n) n
-- Reinhard Zumkeller, Nov 22 2014

[1] cat [(n^(n+1)-n)/(n-1): n in [2..20]]; // Vincenzo Librandi, Apr 16 2015

a:= n-> `if`(n<2, n, (n^(n+1)-n)/(n-1)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2013

f[n_]:=Sum[n^k,{k,n}];Array[f,30] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)

a(1)=1; for n!=1 a(n) = (n^(n+1)-1)/(n-1) - 1. - Benoit Cloitre, Aug 17 2002
a(n) = A031973(n)-1 for n>0. - Robert G. Wilson v, Apr 15 2015
a(n) = n*A023037(n) = n^n - 1 + A023037(n). - Mathew Englander, Oct 20 2020

a(0)=0 prepended by Alois P. Heinz, Oct 22 2019

A226238 a(n) = (n^n - n)/(n - 1).

Original entry on oeis.org

2, 12, 84, 780, 9330, 137256, 2396744, 48427560, 1111111110, 28531167060, 810554586204, 25239592216020, 854769755812154, 31278135027204240, 1229782938247303440, 51702516367896047760, 2314494592664502210318, 109912203092239643840220

Offset: 2

Ralf Stephan, Aug 25 2013

a(n) expressed in base n is written with (n-1) ones followed by a zero. - Michel Marcus, Aug 25 2013

Michael De Vlieger, Table of n, a(n) for n = 2..387
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023. See p. 9.

A diagonal of A228275.

Array[(#^# - #)/(# - 1) &, 18, 2] (* Michael De Vlieger, May 24 2023 *)

PARI
```
a(n)=(n^n-n)/(n-1)
```

def A226238(n): return (n**n-n)//(n-1) # Chai Wah Wu, Sep 28 2023

a(n) = Sum_{k=1..n-1} n^k.

a(n) = A023037(n) - 1, for n>1. - Michel Marcus, Aug 25 2013

A068475 a(n) = Sum_{m=0..n} m*n^(m-1).

Original entry on oeis.org

0, 1, 5, 34, 313, 3711, 54121, 937924, 18831569, 429794605, 10987654321, 310989720966, 9652968253897, 326011399456939, 11901025061692313, 466937872906120456, 19594541482740368161, 875711370981239308953, 41524755927216069067489, 2082225625247428808306410

Offset: 0

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

The closed form comes from taking the derivative of the closed form of A031972, for which each term of this sequence is a derivative. - Jonas Whidden, Oct 18 2011

			a(2) = Sum_{m = 1..2} m*2^(m-1) = 1 + 2*2 = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Derivative sequence of A031972.

Cf. A023037, A062805, A062806, A368534.

a068475 n = sum $ zipWith (*) [1..n] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014

[0] cat [(&+[m*n^(m-1): m in [0..n]]): n in [1..30]]; // G. C. Greubel, Oct 13 2018

Maple
```
a := n->sum(m*n^(m-1),m=1..n);
```

Join[{0}, Table[Sum[m*n^(m-1), {m,0,n}], {n,1,30}]] (* G. C. Greubel, Oct 13 2018 *)

for(n=0,30, print1(if(n==0, 0, sum(m=0,n, m*n^(m-1))), ", ")) \\ G. C. Greubel, Oct 13 2018

a(1) = 1. For n > 1, a(n) = ((n-1)*(n+1)*n^n - n^(n+1) + 1)/(n-1)^2. - Jonas Whidden, Oct 18 2011
a(n) = A062806(n) / n for n>=1. - Reinhard Zumkeller, Nov 22 2014
a(n) = [x^(n-1)] 1/((1 - x)*(1 - n*x)^2). - Peter Bala, Feb 12 2024

A191690 a(n) = n^n-n^(n-1)-n^(n-2)-...-n^2-n-1.

Original entry on oeis.org

0, 1, 14, 171, 2344, 37325, 686286, 14380471, 338992928, 8888888889, 256780503550, 8105545862051, 277635514376232, 10257237069745861, 406615755353655134, 17216961135462248175, 775537745518440716416, 37031913482632035365105

Offset: 1

Juri-Stepan Gerasimov, Jun 11 2011

nonn

			a(1)=0 (=1^1-1), a(2)=1 (=2^2-2-1), a(3)=14 (=3^3-3^2-3-1), a(4)=171 (=4^4-4^3-4^2-4-1).

Danny Rorabaugh, Table of n, a(n) for n = 1..350

Cf. A000312, A023037, A191686.

A191690 := proc(n) n^n-add( n^j,j=0..n-1) ; end proc: # R. J. Mathar, Jun 23 2011

Table[Total[-n^Range[0,n-1]]+n^n,{n,2,20}] (* Harvey P. Dale, Jul 06 2011 *)
f[n_] := ((n - 2) n^n + n)/(n - 1) - 1; f[1] = 0; Array[f, 18] (* Robert G. Wilson v, Apr 16 2015 *)

a(n) = n^n - sum(k=0, n-1, n^k); \\ Michel Marcus, Apr 16 2015

[n^n - sum([n^k for k in range(n)]) for n in range(1,19)] # Danny Rorabaugh, Apr 20 2015

a(n) = A117667(n)-1. - Robert G. Wilson v, Apr 16 2015

a(n) = n^n - A023037(n). - Danny Rorabaugh, Apr 20 2015

A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

Original entry on oeis.org

3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Peter Luschny, Mar 03 2018

nonn

Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.

An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.

			Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1,         (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1),     (A299795) (x = 2, y = 2),
(3^p - 1)/2,     (A003462) (x = 1, y = 3),
3^p - 2^p,       (A135171) (x = 2, y = 3),
p*3^(p - 1),     (A027471) (x = 3, y = 3),
(4^p - 1)/3,     (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p,       (A005061) (x = 3, y = 4),
p*4^(p - 1),     (A002697) (x = 4, y = 4),
(p^p-1)/(p-1),   (A023037),
p^p,             (A000312, A051674).
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.

Peter Luschny, Table of n, a(n) for n = 1..10000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Indices of the nonzero values of A300333.

Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.

using Primes
function isA300332(n)
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 2
    while k <= K
        if k == 7
            K = Int(floor(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
            n == r && return true
        end
        k = nextprime(k+1)
    end
    return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))

A345030 a(n) = Sum_{k=1..n} n^(floor(n/k) - 1).

Original entry on oeis.org

1, 3, 11, 70, 633, 7821, 117709, 2097684, 43047545, 1000010125, 25937439391, 743008621422, 23298085496173, 793714780786669, 29192926036832363, 1152921504875352376, 48661191876077295937, 2185911559749718388655, 104127350297928227579629

Offset: 1

Seiichi Manyama, Jun 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Diagonal of A345032.

Cf. A023037, A332533, A344551, A345036.

a[n_] := Sum[n^(Floor[n/k] - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Jun 06 2021 *)

PARI
```
a(n) = sum(k=1, n, n^(n\k-1));
```

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k * (1 - x^k)/(1 - n*x^k).

A068740 Result after dividing (n^n)! as many times as possible by n!.

Original entry on oeis.org

1, 1, 3, 833712928048000000

Offset: 0

Henry Bottomley, Feb 26 2002

nonn

For prime n, it is also the number of generalized knockout tournament seedings with n players in one match and n rounds (see formula below). - Alexander Karpov, Dec 14 2017
Next term is too large to include.
From Robert G. Wilson v, Dec 14 2017: (Start)
a(4) = 4125147631...      (370 digits)...3291015625,
a(5) = 3483655217...     (7923 digits)...3819109376,
a(6) = 2196422024...   (164237 digits)...0161431552,
a(7) = 4948281440...  (4005981 digits)...0000000000,
a(8) = 4242413765...(102886160 digits)...4619140625,
(End)

			a(3)=833712928048000000 since 3!=6 and (3^3)!=27!=10888869450418352160768000000 which is divisible by 6^13=13060694016 but not 6^14=78364164096.

Robert G. Wilson v, Table of n, a(n) for n = 0..4
Alexander Karpov, Generalized knockout tournaments, National Research University Higher School of Economics. WP7/2017/03.

Cf. A000142, A000312, A023037, A057599, A068741, A068742.

a(n) = A068741(n)/A068742(n).

For p prime, a(p) = (p^p)!/(p!)^((p^p-1)/(p-1)).

A076483 a(n) = n!*Sum_{k=1..n} (k-1)^k/k!.

Original entry on oeis.org

0, 0, 1, 11, 125, 1649, 25519, 458569, 9433353, 219117905, 5677963451, 162457597961, 5087919552253, 173136159558361, 6361282619516343, 250987334850557369, 10584205713321808529, 475079402305823570849, 22614513693572549266291, 1137911105533216112417161

Offset: 0

Henry Bottomley, Oct 14 2002

nonn

Perhaps the largest possible number of ways of choosing (v1, v2, ..., vn), possibly with repetition, from {b1, b2, ..., bn} with b1 < b2 < ... < bn, such that v1 + v2 + ... + vn < b1 + b2 + ... + bn. Clearly the actual number of ways depends on the particular values of {b1, b2, ..., bn}, but {1, n, n^2, ..., n^(n-1)} produces this result for the number of sums strictly less than (n^n-1)/(n-1) = A023037(n).

			a(4) = 4!*(0^1/1! + 1^2/2! + 2^3/3! + 3^4/4!) = 0 + 12 + 32 + 81 = 125.

Seiichi Manyama, Table of n, a(n) for n = 0..386

Row sums of A076482.

Cf. A023037, A075473.

Table[n! Sum[(k-1)^k/k!, {k,n}], {n,0,17}] (* Stefano Spezia, Sep 11 2022 *)

a(n) = n!*sum(k=1, n, (k-1)^k/k!); \\ Seiichi Manyama, Jul 15 2023

Limit_{n->oo} a(n)/(e*a(n-1)) - n = -1/2.

Limit_{n->oo} a(n)/n^n = 1/(e-1).

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)

cp's OEIS Frontend

A088790 Numbers k such that (k^k-1)/(k-1) is prime.

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A031972 a(n) = Sum_{k=1..n} n^k.

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A226238 a(n) = (n^n - n)/(n - 1).

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A068475 a(n) = Sum_{m=0..n} m*n^(m-1).

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A191690 a(n) = n^n-n^(n-1)-n^(n-2)-...-n^2-n-1.

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A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

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Julia

A345030 a(n) = Sum_{k=1..n} n^(floor(n/k) - 1).

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