A002523 -id:A002523 - cp's OEIS frontend

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

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

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8k))/2) and with 0^0 = 1.

Original entry on oeis.org

2, 1, 2, 0, 2, 2, 1, 2, 3, 2, 0, 2, 4, 4, 2, 0, 2, 5, 6, 5, 2, 1, 2, 6, 10, 8, 6, 2, 0, 2, 8, 12, 17, 10, 7, 2, 0, 2, 9, 18, 20, 26, 12, 8, 2, 0, 2, 10, 28, 32, 30, 37, 14, 9, 2, 1, 2, 12, 30, 65, 50, 42, 50, 16, 10, 2, 0, 2, 16, 36, 68, 126, 72, 56, 65, 18, 11, 2, 0, 2, 17, 54, 80, 130

Offset: 0

Henry Bottomley, Jun 22 2000

			a(50) = T(5,4) = 5^2+5^1 = 30

Rows include A010054 (apart from initial term), A007395 and A048645 (offset). Subsequent rows are sums of two powers of a given number and also rewritings of A052216 from a particular base to base 10. Columns include A007395, A000027, A005843, A002522, A002378, A001105, A001093, A034262, A011379, A033431, A002523.

T(n, k) = n^A025581(k)+n^A002262(k)

A193759 Array, by antidiagonals, A(k,n) is the number of prime factors of n^(2^k) + 1, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 3, 1, 2, 0, 1, 2, 2, 1, 2, 1, 0, 1, 2, 2, 2, 3, 1, 3, 0, 1, 2, 2, 2, 3, 2, 2, 2, 0, 1, 2, 6, 2, 4, 3, 3, 2, 2, 0, 1, 3, 5, 2, 4, 3, 3, 3, 3, 1, 3, 0, 1, 4, 7, 3, 4, 3, 4, 3, 2, 2, 2, 1, 0, 1, 5

Offset: 0

Jonathan Vos Post, Aug 11 2011

The main diagonal A(n,n) = number of prime factors of n^(2^n) + 1, counted with multiplicity, begins 0, 1, 1, 3, 2, 4, 3, 6, 6.

			A(4,5) = 3 because 1+5^16 = 152587890626 = 2 * 2593 * 29423041, which has 3 prime factors. The array begins:
================================================================
....|n=0|n=1|n=2|n=3|n=4|n=5|n=6|n=7|n=8|n=9|.10|.11|comment
====|===|===|===|===|===|===|===|===|===|===|===|===|===========
k=0.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.3.|.2.|.2.|.1.|.3.|A001222
k=1.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.3.|.2.|.2.|.1.|.2.|A193330
k=2.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.2.|.2.|.3.|.2.|.2.|A193929
k=3.|.0.|.1.|.1.|.3.|.1.|.3.|.2.|.3.|.3.|.2.|.2.|.3.|A194003
k=4.|.0.|.1.|.1.|.2.|.2.|.3.|.3.|.3.|.3.|.2.|.5.|.3.|not in OEIS
k=5.|.0.|.1.|.2.|.2.|.2.|.4.|.3.|.4.|.3.|.2.|.4.|.4.|not in OEIS
================================================================

Cf. A001222, A002523, A060890, A193330, A193929, A194003.

Edited by Alois P. Heinz, Aug 11 2011

More terms from Max Alekseyev, Sep 09 2011

cp's OEIS Frontend

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

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A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8k))/2) and with 0^0 = 1.

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Formula

A193759 Array, by antidiagonals, A(k,n) is the number of prime factors of n^(2^k) + 1, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)*(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8*k))/2) and with 0^0 = 1.

Views

Author

Keywords

Examples

Crossrefs

Formula

A193759 Array, by antidiagonals, A(k,n) is the number of prime factors of n^(2^k) + 1, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A055378 Table read by antidiagonals: T(n,k) = n^trinv(k)+n^(k-((trinv(k)(trinv(k)-1))/2)) where trinv (k) = floor((1+sqrt(1+8k))/2) and with 0^0 = 1.