A019323 -id:A019323 - cp's OEIS frontend

A085030 Number of prime factors of cyclotomic(n,5), which is A019323(n), the value of the n-th cyclotomic polynomial evaluated at x=5.

2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 1, 4, 2, 4, 1, 3, 2, 2, 3, 2, 2, 4, 2, 3, 3, 2, 2, 2, 4, 2, 2, 3, 2, 1, 1, 1, 2, 3, 3, 3, 4, 5, 1, 1, 2, 2, 2, 3, 3, 3, 4, 3, 2, 5, 4, 4, 2, 2, 3, 3, 3, 5, 2, 3, 2, 3, 2, 2, 4, 3, 2, 3, 4, 3, 1, 6, 1, 2, 1, 4, 3, 4, 2, 3, 3, 4, 3

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057956, number of prime factors of 5^n-1.

See references at A085021

Max Alekseyev, Table of n, a(n) for n = 1..502

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), this sequence (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019323, A057956, A059887, A074479.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 5]]][[2]], {n, 1, 100}]

A138935 Indices k such that A019323(k)=Phi[k](5) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 7, 10, 11, 12, 13, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536, 619, 620, 694, 798, 897, 929, 981, 992, 1064, 1134, 1230, 1670, 1807, 2094, 2369

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Can there be an odd multiple of 5 in this sequence?

Robert Price, Table of n, a(n) for n = 1..110
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019323, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 5]] &]

for( i=1,999, ispseudoprime( polcyclo(i,5)) && print1( i","))

a(48)-a(54) from Robert Price, Apr 14 2012

A019325 Cyclotomic polynomials at x=7.

Original entry on oeis.org

7, 6, 8, 57, 50, 2801, 43, 137257, 2402, 117993, 2101, 329554457, 2353, 16148168401, 102943, 4956001, 5764802, 38771752331201, 117307, 1899815864228857, 5649505, 11898664849, 247165843, 4561457890013486057, 5762401, 79797014141614001, 12111126301, 1628413638264057

Offset: 0

Simon Plouffe

Sequence has a(0) = x; see comments in A020501.

Robert Israel, Table of n, a(n) for n = 0..1186
Index entries for cyclotomic polynomials, values at X

Cf. A019320, A019321, A019322, A019323, A019324.

with(numtheory,cyclotomic); f := n->subs(x=7,cyclotomic(n,x)); seq(f(i),i=0..64);

Join[{7}, Cyclotomic[Range[50], 7]] (* Paolo Xausa, Feb 26 2024 *)

a(n) = if(n==0, 7, polcyclo(n, 7)); \\ Michel Marcus, Dec 16 2017

More terms from Michel Marcus, Dec 17 2017

A093109 Numbers n such that the Zsigmondy number Zs(n,5,1) differs from the n-th cyclotomic polynomial evaluated at 5.

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 32, 42, 52, 54, 55, 64, 93, 128, 162, 171, 256, 272, 294, 355, 406, 486, 506, 512, 605, 676, 820, 1024, 1332, 1458, 1474, 1711, 1806, 1830, 2048, 2058, 2162, 2504, 2525, 2715, 2756, 2883, 2943, 3081, 3249, 3629, 3916, 4096, 4374, 4624, 5210

Offset: 1

Ralf Stephan, Mar 20 2004

nonn

Numbers n such that A019323(n) does not equal A064081(n).
Vladeta Jovovic points out that the sequence seems to contain the powers of two as well as the numbers of the form 2*3^k.
Numbers of the form ord(5,p)*p^k where prime p <> 5 and k > 0. Also numbers n > 0 such that A019323(n) =/= 1 (mod n). Also A019323(n) mod n = gcd(n, A019323(n)) = p. - Thomas Ordowski, Oct 22 2017

Cf. A093106, A093107, A093108, A019323, A064081.

Select[Range[10000], GCD[#, Cyclotomic[#, 5]]!=1 &] (* Emmanuel Vantieghem, Nov 13 2016 *)

More terms from Vladeta Jovovic, Apr 02 2004

Definition corrected by Jerry Metzger, Nov 04 2009

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)

cp's OEIS Frontend

A085030 Number of prime factors of cyclotomic(n,5), which is A019323(n), the value of the n-th cyclotomic polynomial evaluated at x=5.

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A138935 Indices k such that A019323(k)=Phi[k](5) is prime, where Phi is a cyclotomic polynomial.

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A019325 Cyclotomic polynomials at x=7.

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A093109 Numbers n such that the Zsigmondy number Zs(n,5,1) differs from the n-th cyclotomic polynomial evaluated at 5.

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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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