login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A189830 Pairs of numbers i,j ordered by increasing i, such that 2 <= j < i, gcd(i,j)=1 and gcd(Phi_j(i), Phi_i(j))=2*i*j+1, where Phi_k(t) is the k-th cyclotomic polynomial. 0
464, 21, 3313, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The second pair i=3313, j=17 is the only known counterexample to a conjecture of Stephens that gcd(Phi_p(q), Phi_q(p))=1 for every pair of prime numbers (p,q). This is related to a conjecture of Feit-Thompson. See the corresponding wiki page. Up to i=9400 there are no new terms of the sequence.

LINKS

Table of n, a(n) for n=1..4.

MathOverflow, Variant of Stephens result

N. M. Stephens, On the Feit-Thompson Conjecture, Math. Comp. 25 (1971), 625.

Wikipedia, Feit-Thompson conjecture

EXAMPLE

i=a(1)=464 and j=a(2)=21 since i=464 is the smallest positive integer such that gcd(Phi_i(j), Phi_j(i)) = 2*i*j+1 for a positive integer j such that 2 <= j < i and gcd(i,j)=1.

PROG

(PARI)

/* define $cy(m, n) = Phi_m(n)$ the $m$-th cyclotomic polynomial evaluated at $t=n$ */

cy(m, n) = {local(po); po = polcyclo(m, t); subst(po, t, n); }

/* for fixed $m$ compute $cy(m, n)$ */

cy1(n) = {subst(po, t, n); }

/* search from m <a to b possible solutions m, n, using parity of m */

scy1(a, b, fr) =

{local(m, n, r, c, c1, c2, d, g, po, be, t1, t2, t3, st, dd);

st = 1; dd = 1; be = b-a; t1 = gettime();

for(m=a, b, po = polcyclo(m, t); if(m % 2 == 0, st = 2; dd = 3; ,

st = 1; dd = 2; );

forstep(n=dd, m-1, st, g = gcd(n, m);

if(g == 1, c1 = cy1(n); r = 2*m*n+1;

if(c1 % r == 0, c2 = cy(n, m); d = gcd(c2, c1);

if(d == r, print([m, n, r]);

); ); ); );

if(m % fr == 0,

t2 = gettime(); t3 = t2+t1; print([m, t3, ((m-a)/be)*100.0]);

t1 = t3;

); ); }

CROSSREFS

Sequence in context: A108834 A233310 A220716 * A050424 A111697 A116344

Adjacent sequences:  A189827 A189828 A189829 * A189831 A189832 A189833

KEYWORD

nonn,more

AUTHOR

Luis H. Gallardo, Apr 28 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 17 12:55 EDT 2021. Contains 343971 sequences. (Running on oeis4.)