A023037 -id:A023037 - cp's OEIS frontend

A128542 a(n) = ((2n)^(2n) - 1)/((2n+1)*(2n-1)).

0, 1, 17, 1333, 266305, 101010101, 62350352785, 56984650387477, 72340172838076673, 121815504877079063701, 262801002506265664160401, 706890015246831381773595701, 2319540481478754999041880822337, 9120177155862455275254332279111413

Offset: 0

Alexander Adamchuk, May 08 2007

nonn

p divides a(p-1) for prime p>3.

G. C. Greubel, Table of n, a(n) for n = 0..190

Cf. A048861 = n^n - 1.

Cf. A023037, A089815.

Concatenation([0], List([1..20], n-> ((2*n)^(2*n)-1)/(4*n^2 -1) )); # G. C. Greubel, Jul 11 2019

[0] cat [((2*n)^(2*n)-1)/(4*n^2 -1): n in [1..20]]; // G. C. Greubel, Jul 11 2019

Join[{0}, Table[((2n)^(2n)-1)/(4n^2-1),{n,1,20}]]

A128542(n)=((n+=n)^n-1)/(n^2-1) \\ M. F. Hasler, Oct 31 2014

[0]+[((2*n)^(2*n)-1)/(4*n^2 -1) for n in (1..20)] # G. C. Greubel, Jul 11 2019

a(n) = ((2n)^(2n)-1)/((2n+1)*(2n-1)).
a(n) = A048861(2n)/((2n+1)*(2n-1)).
a(n) = A023037(2n)/(2n+1).
a(n) = A089815(2n-2).

a(0)=0 added by M. F. Hasler, Oct 31 2014

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

A328005 Number of distinct coefficients in functional composition of 1 + x + ... + x^(n-1) with itself.

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405, 1459

Offset: 0

Vladimir Reshetnikov, Oct 01 2019

Sum_{i=0..n-1} x^i = (x^n - 1)/(x - 1).

			For n = 4, the composition of 1 + x + x^2 + x^3 with itself is 1 + (1 + x + x^2 + x^3) + (1 + x + x^2 + x^3)^2 + (1 + x + x^2 + x^3)^3 = 4 + 6 x + 10 x^2 + 15 x^3 + 15 x^4 + 14 x^5 + 11 x^6 + 6 x^7 + 3 x^8 + x^9 that has 8 distinct coefficients [1, 3, 4, 6, 10, 11, 14, 15], so a(4) = 8.
The first few polynomials p_n(x) are 0, 1, x + 2, x^4 + 2*x^3 + 4*x^2 + 3*x + 3, ... with p_n(1) = A023037(n), n >= 0.

Wikipedia, Function composition

Cf. A099392, A080827, A023037.

f:= n-> unapply(add(x^j, j=0..n-1), x):
a:= n-> nops({coeffs(expand((f(n)@@2)(x)))} minus {0}):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 01 2019

Table[With[{s = Sum[x^k, {k, 0, n - 1}]}, Length[Union[CoefficientList[Expand[s /. x -> s], x]]]], {n, 0, 53}]

a(n)={my(p=(1-x^n)/(1-x)); #Set(Vec(subst(p,x,p)))} \\ Andrew Howroyd, Oct 01 2019

def A328005(n):
    R. = PolynomialRing(ZZ)
    q = R(sum(x^k for k in range(n)))
    return len(Set(q.substitute(x=q).list()))
print([A328005(n) for n in range(55)]) # Peter Luschny, Oct 02 2019

It appears that a(n) = (2*n^2 + (-1)^n + 3)/4 for n >= 5.

Conjectured g.f.: (x^7 - x^6 - x^5 + 2*x^3 + 1)*x/((x + 1)*(1 - x)^3).

A374173 a(n) is the smallest prime whose base-n representation contains a run of at least n identical digits.

Original entry on oeis.org

3, 13, 683, 3907, 55987, 960803, 19173967, 435848051, 11111111113, 1540683021299, 19453310068921, 328114698808283, 45302797058044219, 469172025408063623, 19676527011956855059, 878942778254232811943, 120353718818554114936591, 109912203092239643840221

Offset: 2

Robert P. P. McKone, Jun 30 2024

a(2) to a(18) are all increasing, but a(19) is smaller than a(18).

a(n) = A023037(n) for n in A088790. - Robert Israel, Dec 31 2024

			a(2) = 3 = 11_2.
a(3) = 13 = 111_3.
a(11) = 1540683021299 = 544444444444_11.
a(18) = 120353718818554114936591 = 3111111111111111111_18.
a(19) = 109912203092239643840221 = 1111111111111111111_19.

Robert Israel, Table of n, a(n) for n = 2..385

Cf. A023037, A088790.

f:= proc(n) local t,Q,i,j;
  t:= (n^n-1)/(n-1);
  if isprime(t) then return t fi;
  for i from 1 to n-1 do
    Q:= select(isprime, [seq(i*t*n+j,j=1..n-1),
         seq(i*n^n+j*t,j=1..n-1)]);
    if Q <> [] then return min(Q) fi;
  od;
  FAIL
end proc:
map(f, [$2..20]); # Robert Israel, Dec 31 2024

d[n_]:=d[n]=Table[Table[m,n],{m,0,n-1}];
dpre[n_]:=Flatten[Table[{m}~Join~#&/@d[n],{m,0,n-1}],1];
dpost[n_]:=Flatten[Table[Map[#~Join~{m}&,d[n]],{m,0,n-1}],1];
dprepost[n_]:=Flatten[Table[Map[{j}~Join~#~Join~{m}&,d[n]],{m,0,n-1},{j,0,n-1}],2];
c[n_]:=c[n]=DeleteDuplicates[Sort[Select[FromDigits[#,n]&/@Join[d[n],dpre[n],dpost[n],dprepost[n]],#>n&]]];
a[n_]:=a[n]=Do[If[PrimeQ[q],Return[q];Break[];],{q,c[n]}];
Table[a[n],{n,2,19}]

A343009 a(n) = (n^(2n)-1)/(n^2-1) for n > 1, a(1) = 1.

Original entry on oeis.org

1, 5, 91, 4369, 406901, 62193781, 14129647351, 4467856773185, 1876182941212489, 1010101010101010101, 678356244890331342611, 555922008415320588345745, 546031727340884622966664381, 633213824057681722185793753109, 856031514432518244055765015738351

Offset: 1

Thomas Ordowski, Apr 02 2021

nonn

Conjecture: for n > 2, a(n) is a Fermat pseudoprime to base n.
If p is an odd prime, then a(p) is a Cipolla pseudoprime to base p.
Is a(m) a Fermat pseudoprime to base m for every composite m?
Amiram Eldar confirmed this up to m = 3800.
From Jianing Song, Aug 28 2022: (Start)
a(n) = Product_{d|(2n),d>2} Phi(d,n), where Phi(n,x) is the d-th cyclotomic polynomial. Note that Phi(n,x) > 1 for x >= 2 unless (n,x) = (1,2): suppose that n >= 3 and x >= 2, then Phi(n,x) = Product_{1<=j<=n,gcd(j,n)=1} (x - exp(2*j*Pi*i/n)) = Product_{1<=j<=n/2,gcd(j,n)=1} (x^2 - 2*cos(2*j*Pi/n)*x + 1) = Product_{1<=j<=n/2,gcd(j,n)=1} ((x - cos(2*j*Pi/n))^2 + (sin(2*j*Pi/n))^2) > 1 since x - cos(2*j*Pi/n) > 1. This shows that a(n) is composite for n > 2.
For n > 2, a(n) is a Fermat pseudoprime to base n, since n^(2*n) == 1 (mod a(n)) and 2*n divides a(n)-1 = n^2*(n^(2*n-2)-1)/(n^2-1): if n is even, then 2*n | n^2; if n is odd, then n | n^2 and 2 | n^2+1 = (n^4-1)/(n^2-1) | (n^(2*n-2)-1)/(n^2-1). (End)

			a(10) = (10^20-1)/99 = 1010101010101010101.

Cf. A005563, A023037, A117812, A124899, A132411, A210461.

a[n_] := (n^(2*n)-1)/(n^2-1); a[1] = 1; Array[a, 15] (* Amiram Eldar, Apr 02 2021 *)

a(n) = Sum_{k=0..n-1} n^(2*k). - Davide Rotondo, Aug 28 2022
From Alois P. Heinz, Aug 28 2022: (Start)
a(n) = A117812(n)/A005563(n-1) = A117812(n)/A132411(n-1) for n>=2.
Limit_{n -> 1} (n^(2*n)-1)/(n^2-1) = 1. (End).

More terms from Amiram Eldar, Apr 02 2021

a(1)=1 prepended and name adapted by Alois P. Heinz, Aug 28 2022

cp's OEIS Frontend

A128542 a(n) = ((2n)^(2n) - 1)/((2n+1)*(2n-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Python

SageMath

A328005 Number of distinct coefficients in functional composition of 1 + x + ... + x^(n-1) with itself.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

SageMath

Formula

A374173 a(n) is the smallest prime whose base-n representation contains a run of at least n identical digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A343009 a(n) = (n^(2n)-1)/(n^2-1) for n > 1, a(1) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions