A060885 -id:A060885 - cp's OEIS frontend

A286301 Primes of the form p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

12207031, 2141993519227, 178250690949465223, 2346320474383711003267, 398341412240537151131351, 79545183674814239059370551, 494424256962371823779424877, 8271964541879648991904246901, 32142180034067960734115528951, 91264002187709396686868598317

Offset: 1

Hartmut F. W. Hoft, May 05 2017

nonn

			Prime number 12207031 = Sum_{i=0..10} 5^i is the first in the sequence since 23 divides 88573 = Sum_{i=0..10} 3^i as well as 2047 = Sum_{i=0..10} 2^i.

Subsequence of A060885, A162861 and A193574.

Cf. A162862, A198244, A240693.

a286301[n_] := Select[Map[(Prime[#]^11-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a286301[150] (* data *)

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)

A198244 Primes of the form k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 where k is nonprime.

Original entry on oeis.org

11, 10778947368421, 17513875027111, 610851724137931, 614910264406779661, 22390512687494871811, 22793803793211153712637, 79905927161140977116221, 184251916941751188170917, 319465039747605973452001, 1311848376806967295019263, 1918542715220370688851293

Offset: 1

Jonathan Vos Post, Dec 21 2012

nonn

Subsequence of A060885.
From Bernard Schott, Nov 01 2019: (Start)
These are the primes associated with the terms k of A308238.
A162861 = A286301 Union {this sequence}.
The numbers of this sequence R_11 = 11111111111_k with k > 1 are Brazilian primes, so belong to A085104. (End)

			10778947368421 is in the sequence since 10778947368421 = 20^10 + 20^9 + 20^8 + 20^7 + 20^6 + 20^5 + 20^4 + 20^3 + 20^2 + 20 + 1, 20 is not prime, and 10778947368421 is prime.

Cf. A162861, A000040, A088548, A192321, A102909, A060885, A308238.

Similar to A185632 (k^2+ ...), A193366 (k^4+ ...), A194194 (k^6+ ...).

[a: n in [0..500] | not IsPrime(n) and IsPrime(a) where a is (n^10+n^9+n^8+n^7+n^6+n^5+n^4+n^3+n^2+n+1)]; // Vincenzo Librandi, Nov 09 2014

f:= proc(n)
local p,j;
if isprime(n) then return NULL fi;
p:= add(n^j,j=0..10);
if isprime(p) then p else NULL fi
end proc:
map(f, [$1..1000]); # Robert Israel, Nov 19 2014

forcomposite(n=0,10^3,my(t=sum(k=0,10,n^k));if(isprime(t),print1(t,", "))); \\ Joerg Arndt, Nov 10 2014

from sympy import isprime
A198244_list, m = [], [3628800, -15966720, 28828800, -27442800, 14707440, -4379760, 665808, -42240, 682, 0, 1]
for n in range(1,10**4):
    for i in range(10):
        m[i+1]+= m[i]
    if not isprime(n) and isprime(m[-1]):
        A198244_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

{A060885(A018252(n)) which are in A000040}.

a(5)-a(6) from Robert G. Wilson v, Dec 21 2012
a(7) from Michael B. Porter, Dec 27 2012
Corrected terms a(6)-a(7) and added terms by Chai Wah Wu, Nov 09 2014

A020519 11th cyclotomic polynomial evaluated at powers of 2.

Original entry on oeis.org

11, 2047, 1398101, 1227133513, 1172812402961, 1162219258676257, 1171221845949812801, 1189887617730934227073, 1213666705181745367548161, 1240362622532514091484054017, 1268889750375080065623288448001, 1298708349570020393652962442872833

Offset: 0

Simon Plouffe

Harvey P. Dale, Table of n, a(n) for n = 0..332
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. Cites this sequence.

Cf. A000079, A060885.

with(numtheory,cyclotomic):seq(cyclotomic(11,2^i),i=0..24);

Table[Total[x^Range[0,10]],{x,2^Range[0,10]}] (* Harvey P. Dale, Mar 05 2014 *)

a(n) = polcyclo(11, 2^n); \\ Michel Marcus, Apr 12 2014

G.f.: -(72022409665839104*x^10 -95936100375199744*x^9 +41035167933923328*x^8 -7266644321632256*x^7 +581441424424960*x^6 -21804053415936*x^5 +388080675136*x^4 -3261182144*x^3 +12564486*x^2 -20470*x +11) / ((x -1)*(2*x -1)*(4*x -1)*(8*x -1)*(16*x -1)*(32*x -1)*(64*x -1)*(128*x -1)*(256*x -1)*(512*x -1)*(1024*x -1)). - Colin Barker, Feb 14 2015
a(n) = 1+2^n+4^n+8^n+16^n+32^n+64^n+128^n+256^n+512^n+1024^n. - Colin Barker, Feb 15 2015
a(n) = A060885(A000079(n)). - Michel Marcus, Apr 06 2016

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

A286301 Primes of the form p^10 + p^9 + p^8 + p^7 + p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

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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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A198244 Primes of the form k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 where k is nonprime.

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A020519 11th cyclotomic polynomial evaluated at powers of 2.

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

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