A053699 -id:A053699 - cp's OEIS frontend

A237445 Primes p such that f(f(p)) is prime, where f(x) = x^4 + x^3 + x^2 + x + 1 = A053699(x).

1451, 2351, 2381, 2791, 5531, 5981, 7841, 8821, 10091, 10501, 11411, 11701, 12011, 13241, 15271, 15331, 16691, 17231, 18341, 18671, 19891, 20981, 21911, 23071, 23131, 23561, 23741, 24061, 25321, 27361, 29221, 30851, 30941, 31271, 32141, 33931

Offset: 1

Derek Orr, Feb 08 2014

nonn

All numbers are congruent to 1 mod 10.

			1451 is prime and f(f(1451)) = 387147304469214558406348338836395337085545589397781 is prime. Thus, 1451 is a member of this sequence.

Cf. A088548, A237361.

f(x)=x^4+x^3+x^2+x+1;forprime(p=1,35000,ispseudoprime(f(f(p)))&&print1(p",")) \\ M. F. Hasler, Feb 09 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**5) if isprime(n) and isprime((n**4+n**3+n**2+n+1)**4+(n**4+n**3+n**2+n+1)**3+(n**4+n**3+n**2+n+1)**2+(n**4+n**3+n**2+n+1)+1)}

A055129 Repunits in different bases: table by antidiagonals of numbers written in base k as a string of n 1's.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 13, 15, 5, 1, 6, 21, 40, 31, 6, 1, 7, 31, 85, 121, 63, 7, 1, 8, 43, 156, 341, 364, 127, 8, 1, 9, 57, 259, 781, 1365, 1093, 255, 9, 1, 10, 73, 400, 1555, 3906, 5461, 3280, 511, 10, 1, 11, 91, 585, 2801, 9331, 19531, 21845, 9841, 1023, 11

Offset: 1

Henry Bottomley, Jun 14 2000

			T(3,5)=31 because 111 base 5 represents 25+5+1=31.
      1       1       1       1       1       1       1
      2       3       4       5       6       7       8
      3       7      13      21      31      43      57
      4      15      40      85     156     259     400
      5      31     121     341     781    1555    2801
      6      63     364    1365    3906    9331   19608
      7     127    1093    5461   19531   55987  137257
Starting with the second column, the q-th column list the numbers that are written as 11...1 in base q. - _John Keith_, Apr 12 2021

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (1 <= n <= 150).

Rows include A000012, A000027, A002061, A053698, A053699, A053700. Columns (see recurrence) include A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002275, A016123, A016125. Diagonals include A023037, A031973. Numbers in the table (apart from the first column and first two rows) are ordered in A053696.

A055129 := proc(n,k)
    add(k^j,j=0..n-1) ;
end proc: # R. J. Mathar, Dec 09 2015

Table[FromDigits[ConstantArray[1, #], k] &[n - k + 1], {n, 11}, {k, n, 1, -1}] // Flatten (* or *)
Table[If[k == 1, n, (k^# - 1)/(k - 1) &[n - k + 1]], {n, 11}, {k, n, 1, -1}] // Flatten (* Michael De Vlieger, Dec 11 2016 *)

T(n, k) = (k^n-1)/(k-1) [with T(n, 1) = n] = T(n-1, k)+k^(n-1) = (k+1)*T(n-1, k)-k*T(n-2, k) [with T(0, k) = 0 and T(1, k) = 1].
From Werner Schulte, Aug 29 2021 and Sep 18 2021: (Start)
T(n,k) = 1 + k * T(n-1,k) for k > 0 and n > 1.
Sum_{m=2..n} T(m-1,k)/Product_{i=2..m} T(i,k) = (1 - 1/Product_{i=2..n} T(i,k))/k for k > 0 and n > 1.
Sum_{n > 1} T(n-1,k)/Product_{i=2..n} T(i,k) = 1/k for k > 0.
Sum_{i=1..n} k^(i-1) / (T(i,k) * T(i+1,k)) = T(n,k) / T(n+1,k) for k > 0 and n > 0. (End)

A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

Original entry on oeis.org

3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 17, 14, 7, 8, 12, 26, 26, 22, 8, 9, 14, 37, 42, 53, 30, 9, 10, 16, 50, 62, 106, 80, 46, 10, 11, 18, 65, 86, 187, 170, 161, 62, 11, 12, 20, 82, 114, 302, 312, 426, 242, 94, 12, 13, 22, 101, 146, 457, 518, 937, 682, 485, 126, 13

Offset: 1

Jason Kimberley, Oct 27 2011

k >= 2; g >= 3.
The base k-1 reading of the base 10 string of A094626(g).
Exoo and Jajcay Theorem 1: M(k,g) <= A054760(k,g) with equality if and only if: k = 2 and g >= 3; g = 3 and k >= 2; g = 4 and k >= 2; g = 5 and k = 2, 3, 7 or possibly 57; or g = 6, 8, or 12, and there exists a symmetric generalized n-gon of order k - 1.

			This is the table formed from the antidiagonals for k+g = 5..20:
3   4   5   6    7    8    9     10    11    12    13    14    15   16  17 18
4   6  10  14   22   30    46    62    94   126   190   254   382  510 766
5   8  17  26   53   80   161   242   485   728  1457  2186  4373 6560
6  10  26  42  106  170   426   682  1706  2730  6826 10922 27306
7  12  37  62  187  312   937  1562  4687  7812 23437 39062
8  14  50  86  302  518  1814  3110 10886 18662 65318
9  16  65 114  457  800  3201  5602 22409 39216
10 18  82 146  658 1170  5266  9362 42130
11 20 101 182  911 1640  8201 14762
12 22 122 222 1222 2222 12222
13 24 145 266 1597 2928
14 26 170 314 2042
15 28 197 366
16 30 226
17 32
18

E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Tokyo, Sect. 1A, 20 (1973) 191-208.
R. M. Damerell, On Moore graphs, Proc. Cambridge Phil. Soc. 74 (1973) 227-236.

Jason Kimberley, Table of n, a(n) for n = 1..20100 (k+g = 5..204)
Jason Kimberley, Table of n, k+g, k, g, M(k,g)=a(n) for k+g = 5..204 (n = 1..20100)
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency

Moore lower bound on the order of a (k,g) cage: this sequence (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7), 2*A053698 (g=8), 2*A053699 (g=10), 2*A053700 (g=12), 2*A053716 (g=14), 2*A053716 (g=16), 2*A102909 (g=18), 2*A103623 (g=20), 2*A060885 (g=22), 2*A105067 (g=24), 2*A060887 (g=26), 2*A104376 (g=28), 2*A104682 (g=30), 2*A105312 (g=32).

Cf. A054760 (the actual order of a (k,g)-cage).

ExtendedStringToInt:=func;
M:=func;
k_:=2;g_:=3;
anti:=func;
[anti(kg):kg in[5..15]];

Table[Function[g, FromDigits[#, k - 1] &@ IntegerDigits@ SeriesCoefficient[x (1 + x)/((1 - x) (1 - 10 x^2)), {x, 0, g}]][n - k + 3], {n, 2, 12}, {k, n, 2, -1}] // Flatten (* Michael De Vlieger, May 15 2017 *)

M(k,2i) = 2 sum_{j=0}^{i-1}(k-1)^j =  string "2"^i read in base k-1.
M(k,2i+1) = (k-1)^i +  2 sum_{j=0}^{i-1}(k-1)^j = string "1"*"2"^i read in base k-1.
Recurrence:
M(k,3) = k + 1,
M(k,2i) = M(k,2i-1) + (k-1)^(i-1),
M(k,2i+1) = M(k,2i) + (k-1)^i.

A060885 a(n) = Sum_{j=0..10} n^j.

Original entry on oeis.org

1, 11, 2047, 88573, 1398101, 12207031, 72559411, 329554457, 1227133513, 3922632451, 11111111111, 28531167061, 67546215517, 149346699503, 311505013051, 617839704241, 1172812402961, 2141993519227, 3780494710543, 6471681049901, 10778947368421, 17513875027111, 27824681019587

Offset: 0

N. J. A. Sloane, May 05 2001

a(n) = Phi_11(n), where Phi_k is the k-th cyclotomic polynomial.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. similar sequences of the type a(n) = Sum_{j=0..m} n^j are:

A000027 (m=1), A002061 (m=2), A053698 (m=3), A053699 (m=4), A053700 (m=5), A053716 (m=6), A053717 (m=7), A102909 (m=8), A103623 (m=9), this sequence (m=10), A105067 (m=11), A060887 (m=12), A104376 (m=13), A104682 (m=14), A105312 (m=15), A269442 (m=16), A269446 (m=18).

[(&+[n^j: j in [0..10]]): n in [0..20]]; // G. C. Greubel, Apr 15 2019

A060885 := proc(n)
        numtheory[cyclotomic](11,n) ;
end proc:
seq(A060885(n),n=0..20) ; # R. J. Mathar, Feb 07 2014

Join[{1},Table[Total[n^Range[0,10]],{n,20}]] (* Harvey P. Dale, Jun 19 2011 *)

a(n) = n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 \\ Harry J. Smith, Jul 14 2009

a(n) = polcyclo(11, n); \\ Michel Marcus, Apr 06 2016

[sum(n^j for j in (0..10)) for n in (0..20)] # G. C. Greubel, Apr 15 2019

a(n) = n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.

G.f.: (1+x^2*(1981+x*(66496+x*(534898+x*(1364848+x*(1233970+ x*(389104+x*(36829+x*(672+x)))))))))/(1-x)^11. - Harvey P. Dale, Jun 19 2011

A056578 a(n) = 1 + 2n + 3n^2 + 4*n^3.

Original entry on oeis.org

1, 10, 49, 142, 313, 586, 985, 1534, 2257, 3178, 4321, 5710, 7369, 9322, 11593, 14206, 17185, 20554, 24337, 28558, 33241, 38410, 44089, 50302, 57073, 64426, 72385, 80974, 90217, 100138, 110761, 122110, 134209, 147082, 160753, 175246, 190585, 206794, 223897, 241918

Offset: 0

Henry Bottomley, Jun 29 2000

			For n>4 this is 4321 translated from base n to base 10.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Note: 1 + 2*x + 3*x^2 + 4*x^3 is the first derivative of 1 + x + x^2 + x^3 + x^4, i.e., A053699.

Cf. A000012, A005408, A056109, A056579.

f[n_]:=1+2*n+3*n^2+4*n^3; lst={}; Do[AppendTo[lst,f[n]],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)

a(n) = (A053699(n+1) - A053699(n-1))/2 - 4*n - 1.
G.f.: (1 + 6*x + 15*x^2 + 2*x^3)/(1-x)^4. - Colin Barker, Jan 10 2012
From Elmo R. Oliveira, Apr 20 2025: (Start)
E.g.f.: exp(x)*(1 + 9*x + 15*x^2 + 4*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Elmo R. Oliveira, Apr 20 2025

A088548 Primes of the form k^4 + k^3 + k^2 + k + 1.

Original entry on oeis.org

5, 31, 2801, 22621, 30941, 88741, 245411, 292561, 346201, 637421, 732541, 837931, 2625641, 3500201, 3835261, 6377551, 15018571, 16007041, 21700501, 28792661, 30397351, 35615581, 39449441, 48037081, 52822061, 78914411, 97039801, 147753211, 189004141, 195534851

Offset: 1

Cino Hilliard, Nov 17 2003

These numbers when >= 31 are primes repunits 11111_n in a base n >= 2, so except 5, they are all Brazilian primes belonging to A085104. (See Links "Les nombres brésiliens", § V.4 - § V.5.) A008858 is generated by the bases n present in A049409. - Bernard Schott, Dec 19 2012

			a(2) = 31 is prime and 31 = 2^4 + 2^3 + 2^2 + 2 + 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A002383, A049409, A085104, A088550.

[a: n in [0..200] | IsPrime(a) where a is n^4+n^3+n^2+n+1]; // Vincenzo Librandi, Jul 16 2012

lst={}; Do[a=1+n+n^2+n^3+n^4; If[PrimeQ[a], AppendTo[lst,a]], {n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 02 2009 *)
Select[Table[n^4+n^3+n^2+n+1, {n,0,2000}], PrimeQ] (* Vincenzo Librandi, Jul 16 2012 *)

polypn(n,p) = { for(x=1,n, if(p%2,y=2,y=1); for(m=1,p, y=y+x^m; ); if(isprime(y),print1(y",")); ) }

from sympy import isprime
print(list(filter(isprime, (k**4+k**3+k**2+k+1 for k in range(120))))) # Michael S. Branicky, May 31 2021

A000040 intersect A053699. - R. J. Mathar, Feb 07 2014

A269442 a(n) = n(n^8 + 1)(n^4 + 1)(n^2 + 1)(n + 1) + 1.

Original entry on oeis.org

1, 17, 131071, 64570081, 5726623061, 190734863281, 3385331888947, 38771752331201, 321685687669321, 2084647712458321, 11111111111111111, 50544702849929377, 201691918794585181, 720867993281778161, 2345488209948553531, 7037580381120954241

Offset: 0

Ilya Gutkovskiy, Feb 26 2016

a(n) = Phi_17(n) where Phi_k(x) is the k-th cyclotomic polynomial.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. similar sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), this sequence (k=17), A060891 (k=18), A269446 (k=19).

List([0..20], n-> n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1); # G. C. Greubel, Apr 24 2019

[n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1: n in [0..20]]; // Vincenzo Librandi, Feb 27 2016

Mathematica
```
Table[Cyclotomic[17, n], {n, 0, 15}]
```

a(n)=n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1 \\ Charles R Greathouse IV, Jul 26 2016

[n*(n^8+1)*(n^4+1)*(n^2+1)*(n+1)+1 for n in (0..20)] # G. C. Greubel, Apr 24 2019

G.f.: (1 +130918*x^2 +62343506*x^3 +4646748160*x^4 +102074708252*x^5 +878064150546*x^6 +3419813860214*x^7 +6502752956958*x^8 +6232856389160*x^9 +3004612851498*x^10 +701875014878*x^11 +73106078368*x^12 +2893069436*x^13 +31542430*x^14 +43674*x^15 +x^16)/(1 - x)^17.

Sum_{n>=0} 1/a(n) = 1.05883117453...

A060887 a(n) = Sum_{j=0..12} n^j.

Original entry on oeis.org

1, 13, 8191, 797161, 22369621, 305175781, 2612138803, 16148168401, 78536544841, 317733228541, 1111111111111, 3452271214393, 9726655034461, 25239592216021, 61054982558011, 139013933454241, 300239975158033, 619036127056621, 1224880286215951, 2336276859014281, 4311578947368421

Offset: 0

N. J. A. Sloane, May 05 2001

a(n) = Phi_13(n) where Let Phi_k is the k-th cyclotomic polynomial.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A053698, A053699, A053700, A053716, A053717, A102909, A103623, A060885, A102909, A104376, A104682, A105067.

[(&+[n^j: j in [0..12]]): n in [0..20]]; // G. C. Greubel, Apr 14 2019

A060887 := proc(n)
        numtheory[cyclotomic](13,n) ;
end proc:
seq(A060887(n),n=0..20) ; # R. J. Mathar, Feb 11 2014

Table[1 + Sum[n^j, {j, 1, 12}], {n, 0, 20}] (* G. C. Greubel, Apr 14 2019 *)

a(n) = { n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 } \\ Harry J. Smith, Jul 14 2009

A060887(n)=polcyclo(13,n) \\ M. F. Hasler, Dec 31 2012

[sum(n^j for j in (0..12)) for n in (0..20)] # G. C. Greubel, Apr 14 2019

a(n) = n^12 + n^11 + n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.
G.f.: (x^12 + 2718*x^11 + 363156*x^10 + 8452952*x^9 + 59276439*x^8 + 155812164*x^7 + 167537592*x^6 + 74214648*x^5 + 12642423*x^4 + 691406*x^3 + 8100*x^2 + 1)/(1-x)^13. - Colin Barker, Oct 29 2012
a(n) = (n^13-1)/(n-1) with a(1) = 13 = lim_{x->1} a(x). - M. F. Hasler, Dec 31 2012

Name changed by G. C. Greubel, Apr 14 2019

A100019 a(n) = n^4 + n^3 + n^2.

Original entry on oeis.org

0, 3, 28, 117, 336, 775, 1548, 2793, 4672, 7371, 11100, 16093, 22608, 30927, 41356, 54225, 69888, 88723, 111132, 137541, 168400, 204183, 245388, 292537, 346176, 406875, 475228, 551853, 637392, 732511, 837900, 954273, 1082368, 1222947

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 19 2004

a(n) are the numbers m such that: j^2 = j + m + sqrt(j*m) with corresponding numbers j given by A002061(n+1), and with sqrt(j*m) = A027444(n) = n* A002061(n+1). - Richard R. Forberg, Sep 03 2013.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000583, A002523, A091940, A071253, A059826, A027445, A053699.

[n^4+n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

A100019:=n->n^4+n^3+n^2: seq(A100019(n), n=0..50); # Wesley Ivan Hurt, Apr 14 2017

f[n_]:=n^4+n^3+n^2;Array[f,50,0] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)

a(n) = n^4 + n^3 + n^2 \\ Indranil Ghosh, Apr 15 2017

From Indranil Ghosh, Apr 15 2017: (Start)
G.f.: -x(3 + 13x + 7x^2 + x^3)/(x - 1)^5
E.g.f.: exp(x)*x*(3 + 11x + 7x^2 + x^3)
(End)

A131992 a(n) = 1 + prime(n) + prime(n)^2 + prime(n)^3 + prime(n)^4.

Original entry on oeis.org

31, 121, 781, 2801, 16105, 30941, 88741, 137561, 292561, 732541, 954305, 1926221, 2896405, 3500201, 4985761, 8042221, 12326281, 14076605, 20456441, 25774705, 28792661, 39449441, 48037081, 63455221, 89451461, 105101005, 113654321

Offset: 1

Reinhard Zumkeller, Aug 06 2007

Thébault shows that a(2) = 121 is the only square in this sequence. - Charles R Greathouse IV, Jul 23 2013

Giovanni Resta has found that 28792661 is the first Sophie Germain prime of this form (and actually of the form p = (n^m-1)/(n-1) for any p-1 > n, m > 1). - M. F. Hasler, Mar 03 2020

			a(1) = 31 because prime(1) = 2 and 1 + 2 + 2^2 + 2^3 + 2^4 = 1 + 2 + 4 + 8 + 16 = 31.

Victor Thébault, Curiosités arithmétiques, Mathesis 62 (1953), pp. 120-129.

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A030514, A008864, A060800, A131993.

Equals A053699 restricted to prime indices. Subsequence of primes is A190527.

Table[Sum[Prime[n]^k, {k, 0, 4}], {n, 30}] (* Alonso del Arte, May 24 2015 *)

a(n)=sigma(prime(n)^4)  \\ Charles R Greathouse IV, Jul 23 2013

a(n) = 1 + A131991(n)*A000040(n).
a(n) = (A050997(n) - 1)/A006093(n).
a(n) = A000203(prime(n)^4). - R. J. Mathar, Mar 15 2018
a(n) = (prime(n)^5 - 1)/(prime(n) - 1) = A053699(prime(n)). (This is also meant by the 2nd formula.) - M. F. Hasler, Mar 03 2020

cp's OEIS Frontend

A237445 Primes p such that f(f(p)) is prime, where f(x) = x^4 + x^3 + x^2 + x + 1 = A053699(x).

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A055129 Repunits in different bases: table by antidiagonals of numbers written in base k as a string of n 1's.

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A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

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A060885 a(n) = Sum_{j=0..10} n^j.

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A056578 a(n) = 1 + 2*n + 3*n^2 + 4*n^3.

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A088548 Primes of the form k^4 + k^3 + k^2 + k + 1.

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A269442 a(n) = n*(n^8 + 1)*(n^4 + 1)*(n^2 + 1)*(n + 1) + 1.

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A056578 a(n) = 1 + 2n + 3n^2 + 4*n^3.

A269442 a(n) = n(n^8 + 1)(n^4 + 1)(n^2 + 1)(n + 1) + 1.