A019326 -id:A019326 - cp's OEIS frontend

A085033 Number of prime factors of cyclotomic(n,8), which is A019326(n), the value of the n-th cyclotomic polynomial evaluated at x=8.

1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 3, 2, 3, 2, 2, 3, 4, 2, 3, 4, 2, 3, 3, 2, 4, 2, 3, 4, 5, 1, 2, 3, 3, 4, 5, 2, 5, 3, 4, 2, 4, 1, 4, 4, 3, 3, 5, 2, 3, 3, 2, 8, 7, 4, 4, 3, 2, 3, 5, 3, 4, 3, 2, 3, 2, 2, 5, 7, 4, 5, 6, 2, 6, 5, 4, 6, 3, 1, 7, 3, 4, 5, 4, 2

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057953, number of prime factors of 8^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..500

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), this sequence (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019326, A057953, A059890, A274908.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 8]]][[2]], {n, 1, 100}]

A138938 Indices k such that A019326(k)=Phi[k](8) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

1, 3, 9, 30, 42, 78, 87, 138, 189, 303, 318, 330, 408, 462, 504, 561, 1002, 1389, 1746, 1794, 2040, 2418, 2790, 3894, 4077, 4722, 6738, 10641, 14610, 14616, 15294, 16662, 18966, 19059, 26142, 27144, 28299, 31638, 33639, 39360

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Cf. A019326, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 8]] &]

for( i=1,999, ispseudoprime( polcyclo(i,8)) && print1( i",")) /* use ...subst( polcyclo(i),x,8)... in PARI < 2.4.2 */

a(17)-a(40) from Robert Price, Apr 20 2012

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)

A259308 a(n) = 1 + sigma(n)^4.

Original entry on oeis.org

2, 82, 257, 2402, 1297, 20737, 4097, 50626, 28562, 104977, 20737, 614657, 38417, 331777, 331777, 923522, 104977, 2313442, 160001, 3111697, 1048577, 1679617, 331777, 12960001, 923522, 3111697, 2560001, 9834497, 810001, 26873857, 1048577, 15752962, 5308417

Offset: 1

Robert Price, Jun 24 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. A259309 (indices of primes in this sequence), A259310 (corresponding primes).

[(1 + SumOfDivisors(n)^4): n in [1..50]]; // Vincenzo Librandi, Jun 24 2015

with(numtheory): A259308:=n->1+sigma(n)^4: seq(A259308(n), n=1..50); # Wesley Ivan Hurt, Jul 09 2015

Table[1 + DivisorSigma[1, n]^4, {n, 10000}]
Table[Cyclotomic[8, DivisorSigma[1, n]], {n, 10000}]

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

a(n) = A019326(A000203(n)). - Michel Marcus, Jun 24 2015

A217611 Primes p such that the octal expansion of 1/p has a unique period length.

Original entry on oeis.org

3, 7, 19, 73, 87211, 262657, 18837001, 77158673929, 5302306226370307681801, 19177458387940268116349766612211, 6113142872404227834840443898241613032969, 328017025014102923449988663752960080886511412965881

Offset: 1

Arkadiusz Wesolowski, Oct 08 2012

Also called generalized unique primes in base 8.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..27
C. K. Caldwell, "Top Twenty" page, Generalized Unique
Wikipedia, Octal

Cf. A019326, A040017 (unique-period primes in base 10).

lst = {}; Do[c = Cyclotomic[n, 8]; q = c/GCD[n, c]; If[PrimePowerQ[q], p = FactorInteger[q][[1, 1]]; AppendTo[lst, p]], {n, 138}]; Sort[lst]

cp's OEIS Frontend

A085033 Number of prime factors of cyclotomic(n,8), which is A019326(n), the value of the n-th cyclotomic polynomial evaluated at x=8.

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A138938 Indices k such that A019326(k)=Phi[k](8) is prime, where Phi is a cyclotomic polynomial.

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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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A259308 a(n) = 1 + sigma(n)^4.

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A217611 Primes p such that the octal expansion of 1/p has a unique period length.

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