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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

%I #131 Sep 20 2021 22:14:07

%S 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,

%T 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,

%U 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

%N Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

%C Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.

%C 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.)

%C 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)).

%H Eric Chen, <a href="/A253240/b253240.txt">Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic polynomial</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic polynomial</a>

%H <a href="https://oeis.org/index/Cy#CyclotomicPolynomialsValuesAtX">Index entries for Cyclotomic polynomials, values at X</a>

%F T(m, n) = Phi_m(n)

%e Read by antidiagonals:

%e m\n 0 1 2 3 4 5 6 7 8 9 10 11 12

%e ------------------------------------------------------

%e 0 1 1 1 1 1 1 1 1 1 1 1 1 1

%e 1 -1 0 1 2 3 4 5 6 7 8 9 10 11

%e 2 1 2 3 4 5 6 7 8 9 10 11 12 13

%e 3 1 3 7 13 21 31 43 57 73 91 111 133 157

%e 4 1 2 5 10 17 26 37 50 65 82 101 122 145

%e 5 1 5 31 121 341 781 ... ... ... ... ... ... ...

%e 6 1 1 3 7 13 21 31 43 57 73 91 111 133

%e etc.

%e The cyclotomic polynomials are:

%e n n-th cyclotomic polynomial

%e 0 1

%e 1 x-1

%e 2 x+1

%e 3 x^2+x+1

%e 4 x^2+1

%e 5 x^4+x^3+x^2+x+1

%e 6 x^2-x+1

%e ...

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

%o (PARI) t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)

%o t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)

%o T(m, n) = if(m==0, 1, polcyclo(m, n))

%o a(n) = T(t1(n), t2(n))

%Y Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.

%Y Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.

%Y Main diagonal is A070518.

%Y 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.

%Y Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.

%Y Indices of primes in main diagonal is A070519.

%Y 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).

%Y Cf. A206942 (all terms (sorted) for rows>2 and columns>1).

%Y Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

%K sign,easy,tabl,nice

%O 0,9

%A _Eric Chen_, Apr 22 2015

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)