cp's OEIS Frontend

A159909 Number of pairs (p,q) of odd primes p < q < r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value.

%I A159909 #21 Feb 16 2025 08:33:10
%S A159909 0,0,0,0,1,0,1,2,3,3,4,2,7,1,3,2,6,6,4,7,9,6,5,10,7,9,8,6,13,9,4,14,
%T A159909 10,10,18,6,12,12,10,16,15,11,18,14,11,19,16,13,19,14,17,22,18,16,17,
%U A159909 18,19,20,19,22,17,19,17,19,19,19,31,25,13,38,20,23,25,23,31,30,31,19
%N A159909 Number of pairs (p,q) of odd primes p < q < r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value.
%C A159909 The cyclotomic polynomial Phi[pqr] can only have coefficients with absolute value > 1 if p,q,r are distinct odd primes, that's why we require 2 < p < q < r. If any of these inequalities is replaced by equality, then Phi[pqr] necessarily has only zero or unit (+-1) coefficients. Sequence A159908 counts all possibilities including these trivial cases.
%H A159909 Robin Visser, <a href="/A159909/b159909.txt">Table of n, a(n) for n = 1..130</a>
%H A159909 Phil Carmody, <a href="http://groups.yahoo.com/group/primenumbers/message/20245">"Cyclotomic polynomial puzzles", in: "primenumbers" group</a>, May 9, 2009.
%H A159909 Phil Carmody, David Broadhurst, Maximilian Hasler, Makoto Kamada, <a href="/A159908/a159908.txt">Cyclotomic polynomial puzzles</a>, digest of 43 messages in primenumbers Yahoo group, May 9, 2009 - May 23, 2013.
%H A159909 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>
%e A159909 a(5)=1 is the first nonzero term, since the smallest example for Phi(pqr) having no coefficient > 1 (in abs. value) for odd primes p<q<r is obtained for r=prime(5), namely Phi(3*7*11).
%o A159909 (PARI) A159909(n) = sum( i=2,n-1, my(pq=prime(n)*prime(i)); sum( j=2,i-1, vecmax(abs(Vec(polcyclo(prime(j)*pq))))==1 ))
%Y A159909 Cf. A117223. [_T. D. Noe_, May 11 2009]
%K A159909 nonn,hard
%O A159909 1,8
%A A159909 _M. F. Hasler_, May 09 2009
%E A159909 Extended by _T. D. Noe_, May 11 2009
%E A159909 More terms from _Robin Visser_, Aug 09 2023