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A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n. 3
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 Weisstein's World of Mathematics, Cyclotomic polynomial
FORMULA
T(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
Main diagonal is A070518.
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).
Sequence in context: A316149 A047071 A124287 * A290472 A060240 A153734
KEYWORD
sign,easy,tabl,nice
AUTHOR
Eric Chen, Apr 22 2015
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)