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A253240 Square array read by antidiagonals: a(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

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)).

LINKS

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

FORMULA

a(m, n) = Phi_m(n)

EXAMPLE

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

...

MATHEMATICA

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

PROG

(PARI) 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))

CROSSREFS

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 numbers in this table (sorted) apart from row 0, 1, 2 and column 0, 1).

Cf. A206864 (all primes in this table (sorted) apart from row 0, 1, 2 and column 0, 1).

Cf. A013595, A051664.

Sequence in context: A316149 A047071 A124287 * A290472 A060240 A153734

Adjacent sequences:  A253237 A253238 A253239 * A253241 A253242 A253243

KEYWORD

sign,easy,tabl,nice

AUTHOR

Eric Chen, Apr 22 2015

STATUS

approved

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Last modified November 16 06:50 EST 2018. Contains 317258 sequences. (Running on oeis4.)