A060884 -id:A060884 - cp's OEIS frontend

A259410 a(n) = 1 - sigma(n) + sigma(n)^2 - sigma(n)^3 + sigma(n)^4.

1, 61, 205, 2101, 1111, 19141, 3641, 47461, 26521, 99451, 19141, 593461, 35855, 318505, 318505, 894661, 99451, 2255605, 152381, 3039331, 1016801, 1634221, 318505, 12747541, 894661, 3039331, 2497561, 9661961, 783871, 26505721, 1016801, 15506821, 5200081

Offset: 1

Robert Price, Jun 26 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. A259411 (indices of primes in this sequence), A259412 (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 27 2015

Table[1 - DivisorSigma[1, n] + DivisorSigma[1, n]^2 - DivisorSigma[1, n]^3 + DivisorSigma[1, n]^4, {n, 1, 10000}]
Table[Cyclotomic[10, DivisorSigma[1, n]], {n, 1, 10000}]

a(n) = polcyclo(10, sigma(n)) \\ Michel Marcus, Jun 26 2015

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

a(n) = A060884(A000203(n)). - Michel Marcus, Jun 26 2015

A020518 10th cyclotomic polynomial evaluated at powers of 2.

Original entry on oeis.org

1, 11, 205, 3641, 61681, 1016801, 16519105, 266354561, 4278255361, 68585520641, 1098438933505, 17583600302081, 281406274007041, 4503049938657281, 72053196259835905, 1152886321308467201, 18446462603027742721, 295145653396718878721, 4722348468539854946305

Offset: 0

Simon Plouffe

Colin Barker, Table of n, a(n) for n = 0..830
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1.
Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).

Cf. A000079, A060884.

with(numtheory,cyclotomic):seq(cyclotomic(10,2^i),i=0..24);

Cyclotomic[10, 2^Range[0, 20]] (* Paolo Xausa, Sep 16 2024 *)

a(n) = polcyclo(10, 2^n) \\ Colin Barker, Feb 14 2015

G.f.: -(704*x^4-544*x^3+174*x^2-20*x+1) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)). - Colin Barker, Feb 14 2015
a(n) = 1-2^n+4^n-8^n+16^n. - Colin Barker, Feb 15 2015
a(n) = A060884(A000079(n)). - Michel Marcus, Apr 06 2016

More terms from Colin Barker, Feb 14 2015

A362783 Square array A(n,k) = (n^(2*k + 1) + 1)/(n + 1), n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 7, 1, 1, 1, 43, 61, 13, 1, 1, 1, 171, 547, 205, 21, 1, 1, 1, 683, 4921, 3277, 521, 31, 1, 1, 1, 2731, 44287, 52429, 13021, 1111, 43, 1, 1, 1, 10923, 398581, 838861, 325521, 39991, 2101, 57, 1, 1, 1, 43691, 3587227, 13421773, 8138021, 1439671

Offset: 0

Juri-Stepan Gerasimov, May 03 2023

			Array begins:
=====================================================================
n/k |  0    1      2       3         4          5            6   ...
----+----------------------------------------------------------------
0   |  1    1      1       1         1          1            1   ...
1   |  1    1      1       1         1          1            1   ...
2   |  1    3     11      43       171        683         2731   ...
3   |  1    7     61     547      4921      44287       398581   ...
4   |  1   13    205    3277     52429     838861     13421773   ...
5   |  1   21    521   13021    325521    8138021    203450521   ...
6   |  1   31   1111   39991   1439671   51828151   1865813431   ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=0..3 are A000012, A002061, A060884, A060888.
Rows n=2..4 are A007583, A066443, A299960.
Main diagonal is A179897.

/* as array */ [[&+[(-n)^j: j in [0..2*k]]: k in [0..6]]: n in [0..6]]; // Juri-Stepan Gerasimov, May 06 2023

A(n,k) = (n^(2*k + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023

A(n,k) = Sum_{j=0..2*k} (-n)^j.

a(49) corrected by Andrew Howroyd, Jan 20 2024

A191012 a(n) = n^5 - n^4 + n^3 - n^2 + n.

Original entry on oeis.org

0, 1, 22, 183, 820, 2605, 6666, 14707, 29128, 53145, 90910, 147631, 229692, 344773, 501970, 711915, 986896, 1340977, 1790118, 2352295, 3047620, 3898461, 4929562, 6168163, 7644120, 9390025, 11441326, 13836447, 16616908, 19827445

Offset: 0

Franz Vrabec, Jun 16 2011

n such that x^5 + x^4 + x^3 + x^2 + x + n factors over the integers.

			a(2) = 22 is in the sequence, because x^5 + x^4 + x^3 + x^2 + x + 22 = (x+2)*(x^4 - x^3 + 3*x^2 - 5*x + 11).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A060884.

[n^5 - n^4 + n^3 - n^2 + n: n in [0..30]]; // Vincenzo Librandi, Jun 18 2011

Maple
```
[seq(n*(n^4-n^3+n^2-n+1),n=0..25)];
```

a(n)=((((n-1)*n+1)*n-1)*n+1)*n \\ Charles R Greathouse IV, Jun 17 2011

a(n) = n*A060884(n).

G.f.: x*(5*x^4 + 32*x^3 + 66*x^2 + 16*x + 1)/(1-x)^6.

A260077 Cyclotomic polynomial value Phi(10,n!).

Original entry on oeis.org

1, 1, 11, 1111, 318505, 205646281, 268365829681, 645113283892561, 2642842746670654081, 17340073528178593019521, 173401165343014841913811201, 2538767097801590027098850323201, 52643875748950481516906361827558401, 1503561738163266360299131304568093619201

Offset: 0

Robert Price, Aug 29 2015

nonn

Robert Price, Table of n, a(n) for n = 0..100
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A060884, A200906, A259264, A259265.

Mathematica
```
Table[Cyclotomic[10, n!], {n, 0, 200}]
```

a(n) = A060884(n!) for n>0.

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

cp's OEIS Frontend

A259410 a(n) = 1 - sigma(n) + sigma(n)^2 - sigma(n)^3 + sigma(n)^4.

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A020518 10th cyclotomic polynomial evaluated at powers of 2.

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A362783 Square array A(n,k) = (n^(2*k + 1) + 1)/(n + 1), n >= 0, k >= 0, read by antidiagonals.

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A191012 a(n) = n^5 - n^4 + n^3 - n^2 + n.

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A260077 Cyclotomic polynomial value Phi(10,n!).

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Mathematica

Formula

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

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Magma

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