A117545 -id:A117545 - cp's OEIS frontend

A117544 Least k such that Phi(n,k), the n-th cyclotomic polynomial evaluated at k, is prime.

3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 6, 1, 4, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 2, 2, 14, 3, 1, 2, 10, 2, 1, 2, 1, 25, 11, 2, 1, 5, 1, 6, 30, 11, 1, 7, 7, 2, 5, 7, 1, 3, 1, 2, 3, 1, 2, 9, 1, 85, 2, 3, 1, 3, 1, 16, 59, 7, 2, 2, 1, 2, 1, 61, 1, 7, 2, 2, 8, 5, 1, 2, 11, 4, 2, 6, 44, 4, 1, 2, 63

Offset: 1

T. D. Noe, Mar 28 2006

nonn

Note that a(n)=1 iff n is a power of a prime.
Because every cyclotomic polynomial is irreducible, it is expected that a(n) exists for all n.
Note that if p=Phi(n,k) is prime and n>1, then p==1 (mod k). - Corrected by Robert Israel, Apr 22 2019

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1000 from T. D. Noe)

Cf. A085398, A117545 (least k such that Phi(k, n) is prime), A307687.

f:= proc(n) local C, x, k;
  C:= unapply(numtheory:-cyclotomic(n, x), x);
  for k from 1 do if isprime(C(k)) then return k fi od
end proc:
map(f, [$1..200]); # Robert Israel, Apr 22 2019

Table[k=1; While[ !PrimeQ[Cyclotomic[n,k]], k++ ]; k, {n,100}]

a(n) = my(k=1); while (!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Apr 22 2019

Phi(n, a(n)) = A307687(n). - Robert Israel, Apr 22 2019

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)

A241039 Cyclotomic(n,2048).

Original entry on oeis.org

1, 2047, 2049, 4196353, 4194305, 17600780175361, 4192257, 73823022692637345793, 17592186044417, 73786976303428141057, 17583600302081, 1298708349570020393652962442872833, 17592181850113

Offset: 0

T. D. Noe, Apr 15 2014

nonn

Are all terms composite? At least the first 10000 terms are.

T. D. Noe, Table of n, a(n) for n = 0..300

Cf. A019320-A019331 (cyclotomic polynomials evaluated at 2..13).
Cf. A020500-A020513 (cyclotomic polynomials evaluated at 1, -2..-13, -1).
Cf. A117544 (least k such that cyclotomic(n,k) is prime).
Cf. A117545 (least k such that cyclotomic(k,n) is prime).

Mathematica
```
Table[Cyclotomic[k, 2048], {k, 0, 20}]
```

A250209 a(n) = least k such that k * n is in A072226, or 0 if no such k exists.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 5, 1, 1, 1, 1, 1, 8, 34, 8, 1, 2, 1, 10, 1, 2, 350, 2, 1, 111, 4, 1, 3, 16, 4, 15, 28, 3, 1, 206, 3, 10, 2, 1, 1, 2, 3, 1, 15, 637, 12, 1, 4, 22, 17, 104, 4, 2, 1012, 1, 1

Offset: 1

Eric Chen, Jan 18 2015

nonn

Conjecture: a(n) > 0 for all n.
a(n) is currently unknown for n = 121, 124, 143, 162, 171, 172, 185, 188, 197, 215, ..., for which we have n * a(n) > 130000.
a(121) = (A117545(2048))/11 and they are both currently unknown.
A117545(2^n) = a(A064549(n)).
a(130) = 917, a(144) = 820, a(164) = 720, a(201) = 606. - Max Alekseyev, Dec 04 2024

Eric Chen, Table of n, a(n) for n = 1..120
Eric Chen, Table of n, a(n) for n = 1..300 status

Table[k=1; While[!PrimeQ[Cyclotomic[n*k, 2]], k++]; k, {n, 43}]

a(n) = {k = 1; while (!isprime(polcyclo(k*n, 2)), k++); k;} \\ Michel Marcus, Jan 18 2015

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A117544 Least k such that Phi(n,k), the n-th cyclotomic polynomial evaluated at k, 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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A241039 Cyclotomic(n,2048).

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A250209 a(n) = least k such that k * n is in A072226, or 0 if no such k exists.

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