This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A253240 #132 Feb 16 2025 08:33:24
%S A253240 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 A253240 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 A253240 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 A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.
%C A253240 Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
%C A253240 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 A253240 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 A253240 Eric Chen, <a href="/A253240/b253240.txt">Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)</a>
%H A253240 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic polynomial</a>
%H A253240 Wikipedia, <a href="http://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic polynomial</a>
%H A253240 <a href="https://oeis.org/index/Cy#CyclotomicPolynomialsValuesAtX">Index entries for Cyclotomic polynomials, values at X</a>
%F A253240 T(m, n) = Phi_m(n)
%e A253240 Read by antidiagonals:
%e A253240 m\n 0 1 2 3 4 5 6 7 8 9 10 11 12
%e A253240 ------------------------------------------------------
%e A253240 0 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A253240 1 -1 0 1 2 3 4 5 6 7 8 9 10 11
%e A253240 2 1 2 3 4 5 6 7 8 9 10 11 12 13
%e A253240 3 1 3 7 13 21 31 43 57 73 91 111 133 157
%e A253240 4 1 2 5 10 17 26 37 50 65 82 101 122 145
%e A253240 5 1 5 31 121 341 781 ... ... ... ... ... ... ...
%e A253240 6 1 1 3 7 13 21 31 43 57 73 91 111 133
%e A253240 etc.
%e A253240 The cyclotomic polynomials are:
%e A253240 n n-th cyclotomic polynomial
%e A253240 0 1
%e A253240 1 x-1
%e A253240 2 x+1
%e A253240 3 x^2+x+1
%e A253240 4 x^2+1
%e A253240 5 x^4+x^3+x^2+x+1
%e A253240 6 x^2-x+1
%e A253240 ...
%t A253240 Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]
%o A253240 (PARI) t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
%o A253240 t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
%o A253240 T(m, n) = if(m==0, 1, polcyclo(m, n))
%o A253240 a(n) = T(t1(n), t2(n))
%Y A253240 Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
%Y A253240 Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
%Y A253240 Main diagonal is A070518.
%Y A253240 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 A253240 Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
%Y A253240 Indices of primes in main diagonal is A070519.
%Y A253240 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 A253240 Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
%Y A253240 Cf. A206864 (all primes (sorted) for rows>2 and columns>1).
%K A253240 sign,easy,tabl,nice
%O A253240 0,9
%A A253240 _Eric Chen_, Apr 22 2015