A081216 -id:A081216 - cp's OEIS frontend

A014566 Sierpiński numbers of the first kind: n^n + 1.

2, 2, 5, 28, 257, 3126, 46657, 823544, 16777217, 387420490, 10000000001, 285311670612, 8916100448257, 302875106592254, 11112006825558017, 437893890380859376, 18446744073709551617, 827240261886336764178, 39346408075296537575425, 1978419655660313589123980

Offset: 0

Eric W. Weisstein

Sierpiński primes of the form n^n + 1 are {2,5,257,...} = A121270. The prime p divides a((p-1)/2) for p = {5,7,13,23,29,31,37,47,53,61,71,...} = A003628 Primes congruent to {5, 7} mod 8. p^2 divides a((p-1)/2) for prime p = {29,37,3373,...}. - Alexander Adamchuk, Sep 11 2006

n divides a(n-1) for even n, or 2n divides a(2n-1). a(2n-1)/(2n) = A124899(n) = {1, 7, 521, 102943, 38742049, 23775972551, 21633936185161, 27368368148803711, 45957792327018709121, ...}. 2^n divides a(2^n-1). A014566[2^n - 1] / 2^n = A081216[2^n - 1] = A122000[n] = {1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, ...}. p+1 divides a(p) for prime p. a(p)/(p+1) = A056852[n] = {7, 521, 102943, 23775972551, 21633936185161, ...}. p^2 divides a((p-1)/2) for prime p = {29, 37, 3373} = A121999(n). - Alexander Adamchuk, Nov 12 2006

Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Maohua Le, Primes in the sequences n^n+1 and n^n-1, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, pp. 156-157.
Paulo Ribenboim, The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 74, 1989.

M. F. Hasler, Table of n, a(n) for n = 0..100
Florentin Smarandache, Only Problems, Not Solutions!, Xiquan Publ. Hse., 1990, Problem 17.
Eric Weisstein's World of Mathematics, Sierpiński Number of the First Kind.

Cf. A000312, A048861, A121270, A003628, A122000, A081216, A056852, A121999, A124899, A134883.

a(0) = 2; for n>0 Table[n^n+1,{n,1,20}] (* Alexander Adamchuk, Sep 11 2006 *)

A014566[n]:=if n=0 then 2 else n^n+1$
makelist(A014566[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

A014566(n)=n^n+1 /* M. F. Hasler, Jan 21 2009 */

For n>0, resultant of x^n+1 and nx-1. - Ralf Stephan, Nov 20 2004
E.g.f.: exp(x) + 1/(1+LambertW(-x)). - Vaclav Kotesovec, Dec 20 2014
Sum_{n>=1} 1/a(n) = A134883. - Amiram Eldar, Nov 15 2020

More terms from Erich Friedman

A062160 Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.

Original entry on oeis.org

0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, 1, 1, 0, 1, 0, 3, 2, 1, 0, -1, 1, 5, 7, 3, 1, 0, 1, 0, 11, 20, 13, 4, 1, 0, -1, 1, 21, 61, 51, 21, 5, 1, 0, 1, 0, 43, 182, 205, 104, 31, 6, 1, 0, -1, 1, 85, 547, 819, 521, 185, 43, 7, 1, 0, 1, 0, 171, 1640, 3277, 2604, 1111, 300, 57, 8, 1, 0, -1, 1, 341, 4921, 13107, 13021, 6665, 2101, 455, 73, 9, 1, 0

Offset: 0

Henry Bottomley, Jun 08 2001

For n >= 1, T(n, k) equals the number of walks of length k between any two distinct vertices of the complete graph K_(n+1). - Peter Bala, May 30 2024

			From _Seiichi Manyama_, Apr 12 2019: (Start)
Square array begins:
   0, 1, -1,  1,  -1,    1,    -1,      1, ...
   0, 1,  0,  1,   0,    1,     0,      1, ...
   0, 1,  1,  3,   5,   11,    21,     43, ...
   0, 1,  2,  7,  20,   61,   182,    547, ...
   0, 1,  3, 13,  51,  205,   819,   3277, ...
   0, 1,  4, 21, 104,  521,  2604,  13021, ...
   0, 1,  5, 31, 185, 1111,  6665,  39991, ...
   0, 1,  6, 43, 300, 2101, 14706, 102943, ... (End)

Seiichi Manyama, Antidiagonals n = 0..139, flattened
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592, A015609.
Columns include A000004, A000012, A023443, A002061, A062158, A060884, A062159, A060888.
Related to repunits in negative bases (cf. A055129 for positive bases).
Main diagonal gives A081216.
Cf. A109502.

seq(print(seq((n^k - (-1)^k)/(n+1), k = 0..10)), n = 0..10); # Peter Bala, May 31 2024

T[n_,k_]:=(n^k - (-1)^k)/(n+1); Join[{0},Table[Reverse[Table[T[n-k,k],{k,0,n}]],{n,12}]]//Flatten (* Stefano Spezia, Feb 20 2024 *)

T(n, k) = n^(k-1) - n^(k-2) + n^(k-3) - ... + (-1)^(k-1) = n^(k-1) - T(n, k-1) = n*T(n, k-1) - (-1)^k = (n - 1)*T(n, k-1) + n*T(n, k-2) = round[n^k/(n+1)] for n > 1.
T(n, k) = (-1)^(k+1) * resultant( n*x + 1, (x^k-1)/(x-1) ). - Max Alekseyev, Sep 28 2021
G.f. of row n: x/((1+x) * (1-n*x)). - Seiichi Manyama, Apr 12 2019
E.g.f. of row n: (exp(n*x) - exp(-x))/(n+1). - Stefano Spezia, Feb 20 2024
From Peter Bala, May 31 2024: (Start)
Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = (m + 1)^(n-1) for n >= 1.
Let R(m, x) denote the g.f. of the m-th row of the square array. Then R(m_1, x) o R(m_2, x) = R(m_1 + m_2 + m_1*m_2, x),  where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A109502.
T(m_1 + m_2 + m_1*m_2, k) = Sum_{i = 0..k} Sum_{j = i..k} binomial(k, i)* binomial(k-i, j-i)*T(m_1, j)*T(m_2, k-i).  (End)

A124899 Sierpinski quotient ((2n-1)^(2n-1) + 1)/(2n) = A014566(2n-1)/(2n).

Original entry on oeis.org

1, 7, 521, 102943, 38742049, 23775972551, 21633936185161, 27368368148803711, 45957792327018709121, 98920982783015679456199, 265572137199362841880960201, 870019499993663001431459704607, 3416070845000481662841943594125601

Offset: 1

Alexander Adamchuk, Nov 12 2006

nonn

2n divides Sierpinski number A014566(2n-1).
2^n divides A014566(2^n-1); A014566(2^n - 1) / 2^n = A081216(2^n - 1) = A122000(n) = {1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, ...}.
p+1 divides A014566(p) for prime p; A014566(p)/(p+1) = A056852(n) = {7, 521, 102943, 23775972551, 21633936185161, ...}.
Primes in this sequence are {7, 521, 45957792327018709121}.

Seiichi Manyama, Table of n, a(n) for n = 1..194
Eric Weisstein, World of Mathematics. Sierpinski Numbers of the First Kind.

Cf. A014566 (Sierpinski numbers of the first kind: n^n + 1).

Cf. A056826, A056852, A081216, A122000.

List([1..15],n->((2*n-1)^(2*n-1)+1)/(2*n)); # Muniru A Asiru, Apr 08 2018

seq(((2*n-1)^(2*n-1)+1)/(2*n),n=1..20); # Muniru A Asiru, Apr 08 2018

Mathematica
```
Table[((2n-1)^(2n-1)+1)/(2n),{n,1,20}]
```

a(n) = ((2*n-1)^(2*n-1) + 1)/(2*n); \\ Michel Marcus, Apr 08 2018

a(n) = ((2n-1)^(2n-1) + 1)/(2n) = A014566(2n-1)/(2n).

(2n-1)^(a(n)-1) == 1 (mod a(n)). - Thomas Ordowski, Mar 16 2021

A083062 a(n) = (n+1)^n/(n+2) - (-1)^n/(n+2).

Original entry on oeis.org

0, 1, 2, 13, 104, 1111, 14706, 233017, 4304672, 90909091, 2161452050, 57154490053, 1664148937320, 52914318216943, 1824557876586914, 67818912035696881, 2703399548648159360, 115047976828352449051, 5206367514895562076642, 249660952380952380952381

Offset: 0

Paul Barry, Apr 20 2003

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

Cf. A081216, A081209.

Table[(n+1)^n/(n+2)-(-1)^n/(n+2),{n,0,20}] (* Harvey P. Dale, Oct 08 2013 *)

A122000 a(n) = ((2^n - 1)^(2^n - 1) + 1) / 2^n.

Original entry on oeis.org

1, 7, 102943, 27368368148803711, 533411691585101123706582594658103586126397951, 3566766192921360077810945505268211287512797261288920841093043641769808083046939618603793791988232043305924036607

Offset: 1

Alexander Adamchuk, Sep 11 2006

nonn

A014566(n) = n^n + 1 is Sierpinski Number of the First Kind. A014566(2^n - 1) is divisible by 2^n. a(n) is a subset of A081216(n) = (n^n-(-1)^n)/(n+1).

2^p - 1 divides a(p-1) for prime p>2. Corresponding quotients are a(p-1) / (2^p - 1) = {1, 882850585445281, 28084773172609134470952326813135521948919663474715912134590894817085103016117634792155856629828598766188378241, ...}, where p = prime(n) for n>1. - Alexander Adamchuk, Jan 22 2007

Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind.

Cf. A014566, A081216, A056009.

Mathematica
```
Table[((2^n-1)^(2^n-1)+1)/2^n,{n,1,7}]
```

a(n) = A014566(2^n - 1) / 2^n.
a(n) = A081216(2^n - 1).
a(n) = A056009(2^n - 1).

A083063 a(n) = (n+1)^(n-1)/(n+2) + (-1)^n/(n+2).

Original entry on oeis.org

1, 0, 1, 3, 21, 185, 2101, 29127, 478297, 9090909, 196495641, 4762874171, 128011456717, 3779594158353, 121637191772461, 4238682002231055, 159023502861656433, 6391554268241802725, 274019342889240109297

Offset: 0

Paul Barry, Apr 20 2003

Cf. A083062, A081216, A081209.

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

A128446 Quotients A122000(p-1) / (2^p - 1), where p = prime(n) for n > 1.

Original entry on oeis.org

1, 882850585445281, 28084773172609134470952326813135521948919663474715912134590894817085103016117634792155856629828598766188378241

Offset: 2

Alexander Adamchuk, Mar 03 2007

A014566(n) = n^n + 1 is a Sierpinski Number of the First Kind.
A014566(2^n - 1) is divisible by 2^n.
A122000(n) = ((2^n - 1)^(2^n - 1) + 1) / 2^n = A014566(2^n - 1) / 2^n = A081216(2^n - 1).
a(5) = 6.044...*10^3072, and is too large to include. - Amiram Eldar, Jul 17 2025

Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind.

Cf. A122000, A014566, A081216, A056009.

a[n_] := Module[{p = Prime[n]}, ((2^(p-1) - 1)^(2^(p-1) - 1) + 1)/(2^(p-1)*(2^p-1))]; Array[a, 3, 2] (* Amiram Eldar, Jul 17 2025 *)

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

A280797 a(n) = (n^n - 1)*(n^n + 1)/(n + 1).

Original entry on oeis.org

0, 0, 5, 182, 13107, 1627604, 310968905, 84777884106, 31274997412295, 15009463529699912, 9090909090909090909, 6783562448903313426110, 6115142092568526471803195, 6552380728090615475599972572, 8231779712749862388415318790417, 11984441202055255416780710220336914

Offset: 0

Juri-Stepan Gerasimov, Jan 08 2017

			a(0) = 0 because (0^0 - 1)*(0^0 + 1)/(0 + 1) = 0,
a(1) = 0 because (1^1 - 1)*(1^1 + 1)/(1 + 1) = 0,
a(2) = 5 because (2^2 - 1)*(2^2 + 1)/(2 + 1) = 5.

Cf. A000312, A081216, A117812, A179897.

Magma
```
[(n^(2*n)-1)/(n+1): n in [0..15]];
```

Table[If[n == 0, 0, (n^n - 1) (n^n + 1)/(n + 1)], {n, 0, 15}] (* Michael De Vlieger, Jan 09 2017 *)

a(n) = A117812(n)/(n + 1).

Offset changed to 0 by Georg Fischer, Jul 15 2024

cp's OEIS Frontend

A014566 Sierpiński numbers of the first kind: n^n + 1.

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A062160 Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.

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A124899 Sierpinski quotient ((2n-1)^(2n-1) + 1)/(2n) = A014566(2n-1)/(2n).

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A083062 a(n) = (n+1)^n/(n+2) - (-1)^n/(n+2).

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A122000 a(n) = ((2^n - 1)^(2^n - 1) + 1) / 2^n.

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A083063 a(n) = (n+1)^(n-1)/(n+2) + (-1)^n/(n+2).

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A128446 Quotients A122000(p-1) / (2^p - 1), where p = prime(n) for n > 1.

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A280797 a(n) = (n^n - 1)*(n^n + 1)/(n + 1).

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