A060883 -id:A060883 - cp's OEIS frontend

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

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...

A063784 Primes that are the sum of cubes of divisors of some integer.

Original entry on oeis.org

73, 757, 1772893, 48551233240513, 378890487846991, 3156404483062657, 17390284913300671, 280343912759041771, 319913861581383373, 487014306953858713, 7824668707707203971, 8443914727229480773, 32564717507686012813, 48095468363380957093, 54811417636756749151

Offset: 1

Labos Elemer, Aug 17 2001

nonn

Primes of the form p^6 + p^3 + 1 where p is a prime. - Amiram Eldar, Aug 16 2024

			sigma_3(9) = 1 + 27 + 729 = 757, a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A001158, A023194, A060883, A063783, A066100.

Select[Table[p^6 + p^3 + 1, {p, Prime[Range[500]]}], PrimeQ] (* Amiram Eldar, Aug 16 2024 *)

{ n=0; p=0; for (m=1, 10^9, p=nextprime(p+1); if(isprime(q=p^6 + p^3 + 1), write("b063784.txt", n++, " ", q); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 31 2009

Primes of form p = sigma_3(k).
From Amiram Eldar, Aug 16 2024: (Start)
a(n) = A001158(A063783(n)).
a(n) = A060883(A066100(n)). (End)

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)

A259369 a(n) = 1 + sigma(n)^3 + sigma(n)^6.

Original entry on oeis.org

3, 757, 4161, 117993, 46873, 2987713, 262657, 11394001, 4829007, 34018057, 2987713, 481912257, 7532281, 191116801, 191116801, 887533473, 34018057, 3518803081, 64008001, 5489105833, 1073774593, 2176828993, 191116801, 46656216001, 887533473, 5489105833

Offset: 1

Robert Price, Jun 25 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), A060883 (n^6 + n^3 + 1).

Cf. A259370 (indices of primes in this sequence), A259371 (corresponding primes).

[1+SumOfDivisors(n)^3+ SumOfDivisors(n)^6: n in [1..50]]; // Vincenzo Librandi, Jun 26 2015

with(numtheory): A259369:=n->1+sigma(n)^3+sigma(n)^6: seq(A259369(n), n=1..40); # Wesley Ivan Hurt, Jun 29 2015

Table[1 + DivisorSigma[1, n]^3 + DivisorSigma[1, n]^6, {n, 10000}]
Table[Cyclotomic[9, DivisorSigma[1, n]], {n, 10000}]

a(n) = polcyclo(9, sigma(n)) \\ Michel Marcus, Jun 25 2015

a(n) = 1 + A000203(n)^3 + A000203(n)^6.

a(n) = A060883(A000203(n)). - Michel Marcus, Jun 25 2015

A162601 Primes of form k^6 + k^3 + 1.

Original entry on oeis.org

3, 73, 757, 262657, 1772893, 64008001, 85775383, 308933353, 729027001, 15625125001, 17596420453, 30841155073, 46656216001, 225200075257, 885843322057, 1126163480473, 2565165803383, 4608275809411, 5789338864921, 9685393594633, 16157823282721, 25002115044733

Offset: 1

Daniel Tisdale, Jul 07 2009

Robert Israel, Table of n, a(n) for n = 1..10000
Dan Ismailescu and Yunkyu James Lee, Polynomially growing integer sequences all whose terms are composite, arXiv:2501.04851 [math.NT], 2025. See p. 10.

Cf. A000040, A002061, A060883, A153439.

select(isprime, [seq(k^6+k^3+1,k=1..1000)]); # Robert Israel, Jan 12 2025

A000040 INTERSECT A060883. - R. J. Mathar, May 31 2010

Definition clarified, sequence corrected (757 inserted) and extended by R. J. Mathar, May 31 2010

A260076 Cyclotomic polynomial value Phi(9,n!).

Original entry on oeis.org

3, 3, 73, 46873, 191116801, 2985985728001, 139314069877248001, 16390160963204120064001, 4296582355504685658144768001, 2283380023591730863569702223872001, 2283380023591730815832761109839872000001, 4045146997974190235742912149285516869632000001

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. A060883, A200906, A259264, A259265.

Mathematica
```
Table[Cyclotomic[9, n!], {n, 0, 200}]
```

a(n) = A060883(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

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

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A063784 Primes that are the sum of cubes of divisors of some integer.

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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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A259369 a(n) = 1 + sigma(n)^3 + sigma(n)^6.

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PARI

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A162601 Primes of form k^6 + k^3 + 1.

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Extensions

A260076 Cyclotomic polynomial value Phi(9,n!).

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Formula

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

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