cp's OEIS Frontend

A159908 Number of pairs (p,q) of primes p <= q <= r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value.

%I A159908 #22 Feb 16 2025 08:33:10
%S A159908 1,3,6,9,13,15,19,23,27,30,34,35,43,40,45,47,54,57,58,64,69,69,71,79,
%T A159908 79,84,86,87,97,96,94,107,106,109,120,111,120,123,124,133,135,134,144,
%U A159908 143,143,154,154,154,163,161,167,175,174,175,179,183,187,191,193,199,197,202,203
%N A159908 Number of pairs (p,q) of primes p <= q <= r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value.
%C A159908 The cyclotomic polynomial Phi[pqr] (p,q,r primes) can only have coefficients with absolute value > 1 if p,q,r are distinct odd primes. This sequence also counts the trivial cases where (1): p=2, or (2): p=q, or (3): q=r. The number of these cases is A008486(n-1). Sequence A159909 counts only the nontrivial cases.
%H A159908 Robin Visser, <a href="/A159908/b159908.txt">Table of n, a(n) for n = 1..130</a>
%H A159908 Phil Carmody, <a href="http://groups.yahoo.com/group/primenumbers/message/20245">"Cyclotomic polynomial puzzles", in: "primenumbers" group</a>, May 9, 2009.
%H A159908 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 A159908 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%F A159908 a(n) = A008486(n-1) + A159909(n).
%o A159908 (PARI) A159908(n) = sum( i=1,n, my(pq=prime(n)*prime(i)); sum( j=1,i, vecmax(abs(Vec(polcyclo(prime(j)*pq))))==1 ))
%Y A159908 Cf. A008486, A159909.
%K A159908 hard,nonn
%O A159908 1,2
%A A159908 _M. F. Hasler_, May 09 2009
%E A159908 More terms from _Robin Visser_, Aug 09 2023