A053699 -id:A053699 - cp's OEIS frontend

A193366 Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.

5, 22621, 245411, 346201, 637421, 837931, 2625641, 3835261, 6377551, 15018571, 16007041, 21700501, 30397351, 35615581, 52822061, 78914411, 97039801, 147753211, 189004141, 195534851, 209102521, 223364311, 279086341, 324842131, 421106401, 445120421, 566124791, 693025471, 727832821, 745720141, 880331261, 943280801, 987082981, 1544755411, 1740422941

Offset: 1

Jonathan Vos Post, Dec 20 2012

Note that there are no primes of the form n^3 + n^2 + n + 1 = (n+1)*(n^2+1) as irreducible components over Z.
From Bernard Schott, May 15 2017: (Start)
These are the primes associated with A286094.
A088548 = A190527 Union {This sequence}.
All the numbers of this sequence n^4 + n^3 + n^2 + n + 1 = 11111_n with n > 1 are Brazilian numbers, so belong to A125134 and A085104. (End)

			a(1) = 1^4 + 1^3 + 1^2 + 1 + 1 = 5.
a(2) = 12^4 + 12^3 + 12^2 + 12 + 1 = 22621.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38.

Subsequence of A088548.
Cf. A000040, A018252, A185632, A192321.
Cf. A049409, A053699, A065509, A085104, A088548, A125134, A190527, A286094.

for n from 1 to 150 do p(n):= 1+n+n^2+n^3+n^4;
if tau(n)>2 and isprime(p(n)) then print(n,p(n)) else fi od: # Bernard Schott, May 15 2017

Select[Map[Total[#^Range[0, 4]] &, Select[Range@ 204, ! PrimeQ@ # &]], PrimeQ] (* Michael De Vlieger, May 15 2017 *)

print1(5);forcomposite(n=4,1e3,if(isprime(t=n^4+n^3+n^2+n+1),print1(", "t))) \\ Charles R Greathouse IV, Mar 25 2013

{n^4 + n^3 + n^2 + n + 1 where n is in A018252}.

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A258978 a(n) = 1 + sigma(n) + sigma(n)^2 + sigma(n)^3 + sigma(n)^4.

Original entry on oeis.org

5, 121, 341, 2801, 1555, 22621, 4681, 54241, 30941, 111151, 22621, 637421, 41371, 346201, 346201, 954305, 111151, 2374321, 168421, 3187591, 1082401, 1727605, 346201, 13179661, 954305, 3187591, 2625641, 10013305, 837931, 27252361, 1082401, 16007041, 5421361

Offset: 1

Robert Price, Jun 15 2015

Robert Price, Table of n, a(n) for n = 1..10000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A000203 (sum of divisors of n).

Cf. A258979 (indices of primes in this sequence), A258980 (corresponding primes).

[(1 + DivisorSigma(1, n) + DivisorSigma(1, n)^2 + DivisorSigma(1, n)^3 + DivisorSigma(1, n)^4): n in [1..35]]; // Vincenzo Librandi, Jun 16 2015

with(numtheory): A258978:=n->1+sigma(n)+sigma(n)^2+sigma(n)^3+sigma(n)^4: seq(A258978(n), n=1..40); # Wesley Ivan Hurt, Jul 09 2015

Table[1 + DivisorSigma[1, n] + DivisorSigma[1, n]^2 + DivisorSigma[1, n]^3 + DivisorSigma[1, n]^4, {n, 10000}]
Table[Cyclotomic[5, DivisorSigma[1, n]], {n, 10000}]
Total/@Table[DivisorSigma[1,n]^ex,{n,40},{ex,0,4}] (* Harvey P. Dale, Jun 24 2017 *)

vector(50, n, polcyclo(6, sigma(n))) \\ Michel Marcus, Jun 25 2015

a(n) = 1 + A000203(n) + A000203(n)^2 + A000203(n)^3 + A000203(n)^4.

a(n) = A053699(A000203(n)). - Michel Marcus, Jun 25 2015

A232397 a(n) = ceiling(sqrt(n^4 + n^3 + n^2 + n + 1))^2 - (n^4 + n^3 + n^2 + n + 1).

Original entry on oeis.org

0, 4, 5, 0, 20, 3, 45, 8, 80, 15, 125, 24, 180, 35, 245, 48, 320, 63, 405, 80, 500, 99, 605, 120, 720, 143, 845, 168, 980, 195, 1125, 224, 1280, 255, 1445, 288, 1620, 323, 1805, 360, 2000, 399, 2205, 440, 2420, 483, 2645, 528, 2880, 575, 3125, 624, 3380, 675

Offset: 0

Vladimir Shevelev, Nov 23 2013

a(n) = 0 if and only if n^4 + n^3 + n^2 + n + 1 is a perfect square.
Using formula below, we immediately prove that a(n)=0 iff n=0 or n=3. This means that all nonnegative solutions of the Diophantine equation n^4 + n^3 + n^2 + n + 1 = m^2 are n=0, m=1 and n=3, m=11.
For m >=0, if we also consider negative values of n, we obtain only one more solution: n=-1, m=1.
Indeed, if one considers sequence b(n) = ceiling(sqrt(n^4 - n^3 + n^2 - n + 1))^2 - (n^4 - n^3 + n^2 - n +1 ), then, for even n, a(n) = b(n), while for odd n>=3, a(n) = b(n-2).

Robert Israel, Table of n, a(n) for n = 0..10001
Max Alekseyev, Re: oddness of iterations of sigma(), SeqFan Mailing List.

Cf. A053699, A068527, A232395, A232423.

[Ceiling(Sqrt(n^4+n^3+n^2+n+1))^2-(n^4+n^3+n^2+n+1): n in [0..60]]; // Vincenzo Librandi, Jan 31 2016

0, 4, seq(op([5*k^2, k^2-1]),k=1..100); # Robert Israel, Feb 02 2016

Table[Ceiling[Sqrt[n^4 + n^3 + n^2 + n + 1]]^2 - (n^4 + n^3 + n^2 + n + 1), {n, 0, 60}] (* Vincenzo Librandi, Jan 31 2016 *)

from math import isqrt
def A232397(n): return (1+isqrt(m:=n*(n*(n*(n+1)+1)+1)))**2-m-1 # Chai Wah Wu, Jul 29 2022

a(1) = 4, for other odd n, a(n) = ((n-1)/2)^2 - 1; for even n>=0, a(n) = 5/4 * n^2.
a(n) = A068527(A053699(n)). [Straight from the description: Difference between smallest square >= (n^4 + n^3 + n^2 + n + 1) and (n^4 + n^3 + n^2 + n + 1)]. - Antti Karttunen, Nov 28 2013
a(n) = (6*n^2-2*n-3+(4*n^2+2*n+3)*(-1)^n+20*(1-(-1)^(2^abs(n-1))))/8. - Luce ETIENNE, Jan 30 2016
G.f.: 4*x+x^2*(x^5-3*x^3-5*x^2-5)/(x^2-1)^3. - Robert Israel, Feb 02 2016

More terms from Peter J. C. Moses

A100606 a(n) = n^4 + n^3 + n.

Original entry on oeis.org

0, 3, 26, 111, 324, 755, 1518, 2751, 4616, 7299, 11010, 15983, 22476, 30771, 41174, 54015, 69648, 88451, 110826, 137199, 168020, 203763, 244926, 292031, 345624, 406275, 474578, 551151, 636636, 731699, 837030, 953343, 1081376, 1221891, 1375674, 1543535, 1726308

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 30 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A002523, A027445, A053699, A059826, A071253, A091940, A100019.

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

Table[n^4+n^3+n,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,3,26,111,324},40] (* Harvey P. Dale, Apr 25 2015 *)

a(n)=n^4+n^3+n \\ Charles R Greathouse IV, Oct 21 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=3, a(2)=26, a(3)=111, a(4)=324. - Harvey P. Dale, Apr 25 2015
From Elmo R. Oliveira, Aug 29 2025: (Start)
G.f.: x*(3 + 11*x + 11*x^2 - x^3)/(1-x)^5.
E.g.f.: x*(3 + 10*x + 7*x^2 + x^3)*exp(x). (End)

A237361 Numbers n of the form n = Phi_5(p) (for prime p) such that Phi_5(n) is also prime.

Original entry on oeis.org

4435770414505, 30562950873505, 32152890387805, 60700878873905, 936037312559305, 1279875801783805, 3780430049614405, 6055088920612205, 10370026462436905, 12160851727605005, 16956369914710105, 18746881534017005, 20813869508536105, 30740855019988405

Offset: 1

Derek Orr, Feb 06 2014

nonn

Phi_5(x) = x^4 + x^3 + x^2 + x + 1 is the fifth cyclotomic polynomial, see A053699.
All numbers are congruent to 5 mod 100.
The definition requires p to be prime, Phi_5(p) does not need to be prime, but Phi_5(Phi_5(p)) must be prime.

			4435770414505 = 1451^4+1451^3+1451^2+1451+1 (1451 is prime), and 4435770414505^4+4435770414505^3+4435770414505^2+4435770414505+1 = 387147304469214558406348338836395337085545589397781 is prime. Thus, 4435770414505 is a member of this sequence.

Cf. A131992, A088548.

forprime(p=2,1e7, k=polcyclo(5,p) ; if( ispseudoprime(polcyclo(5,k)), print1(k", "))) \\ Charles R Greathouse IV, Feb 07 2014

import sympy
from sympy import isprime
{print(n**4+n**3+n**2+n+1) 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)}

A326618 a(n) = n^18 + n^9 + 1.

Original entry on oeis.org

1, 3, 262657, 387440173, 68719738881, 3814699218751, 101559966746113, 1628413638264057, 18014398643699713, 150094635684419611, 1000000001000000001, 5559917315850179173, 26623333286045024257, 112455406962561892503, 426878854231297789441, 1477891880073843750001

Offset: 0

Richard N. Smith, Jul 15 2019

nonn

a(n) = Phi_27(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Richard N. Smith, Table of n, a(n) for n = 0..2000
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

Sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=2), A002061 (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), A269442 (k=17), A060891 (k=18), A269446 (k=19), A060892 (k=20), A269483 (k=21), A269486 (k=22), A060893 (k=24), A269527 (k=25), A266229 (k=26), this sequence (k=27), A270204 (k=28), A060894 (k=30), A060895 (k=32), A060896 (k=36).

Cf. A153440 (indices of prime terms).

[n^18+n^9+1: n in [0..17]]; // Vincenzo Librandi, Jul 15 2019

Table[n^18 + n^9 + 1, {n, 0, 17}] (* Vincenzo Librandi, Jul 15 2019 *)
Table[Cyclotomic[27, n], {n, 0, 17}]

a(n) = polcyclo(27, n); \\ Michel Marcus, Jul 20 2019

A342689 Square array read by antidiagonals (upwards): A(n,k) = (k^Fibonacci(n) - 1) / (k - 1) for k >= 0 and n >= 0 with lim_{k -> 1} A(n,k) = A(n,1) = Fibonacci(n).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 5, 7, 4, 1, 1, 0, 1, 8, 31, 13, 5, 1, 1, 0, 1, 13, 255, 121, 21, 6, 1, 1, 0, 1, 21, 8191, 3280, 341, 31, 7, 1, 1, 0, 1, 34, 2097151, 797161, 21845, 781, 43, 8, 1, 1, 0, 1, 55, 17179869184, 5230176601, 22369621, 97656, 1555, 57, 9, 1, 1, 0

Offset: 0

Werner Schulte, May 18 2021

Replacing Fibonacci(n), A000045, with Lucas(n), A000032, you get another square array B(n,k). The terms satisfy the same recurrence equation B(n,k) = (k - 1) * B(n-1,k) * B(n-2,k) + B(n-1,k) + B(n-2,k) for k >= 0 and n > 1 with initial values B(0,k) = k+1 and B(1,k) = 1. Please take account of lim_{k -> 1} (k^Lucas(n) - 1) / (k - 1) = Lucas(n).

			The array A(n,k) for k >= 0 and n >= 0 begins:
n \ k: 0  1           2          3        4     5    6    7  8  9  10  11
=========================================================================
   0 : 0  0           0          0        0     0    0    0  0  0   0   0
   1 : 1  1           1          1        1     1    1    1  1  1   1   1
   2 : 1  1           1          1        1     1    1    1  1  1   1   1
   3 : 1  2           3          4        5     6    7    8  9 10  11  12
   4 : 1  3           7         13       21    31   43   57 73 91 111 133
   5 : 1  5          31        121      341   781 1555 2801
   6 : 1  8         255       3280    21845 97656
   7 : 1 13        8191     797161 22369621
   8 : 1 21     2097151 5230176601
   9 : 1 34 17179869184
  10 : 1 55
  11 : 1 89
  etc.

Cf. A011655 (column k = -1), A057427 (column 0), A000045 (column 1), A063896 (column 2), A000004 (row 0), A000012 (rows 1, 2), A000027 (row 3), A002061 (row 4), A053699 (row 5), A053717 (row 6), A060887 (row 7).

A(n,k) = (k - 1) * A(n-1,k) * A(n-2,k) + A(n-1,k) + A(n-2,k) for k >= 0 and n > 1 with initial values A(0,k) = 0 and A(1,k) = 1.

cp's OEIS Frontend

A193366 Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.

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A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

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A258978 a(n) = 1 + sigma(n) + sigma(n)^2 + sigma(n)^3 + sigma(n)^4.

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A232397 a(n) = ceiling(sqrt(n^4 + n^3 + n^2 + n + 1))^2 - (n^4 + n^3 + n^2 + n + 1).

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A100606 a(n) = n^4 + n^3 + n.

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A237361 Numbers n of the form n = Phi_5(p) (for prime p) such that Phi_5(n) is also prime.

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A326618 a(n) = n^18 + n^9 + 1.

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