A072942 a(n) is the least x such that the cyclotomic polynomial values Phi(d,x) are prime for all d dividing n.
3, 4, 3, 4, 12, 6, 3, 4, 3, 12, 20, 24687390, 3, 72, 62, 4, 20, 1102903830, 12, 58051620, 3, 1793172, 468, 1035844571580, 62, 882, 398, 75274140, 6, 81206805256038, 14, 1288005000, 78428, 93888, 37664, 24380304369772260, 432, 3300, 21962
Offset: 1
Keywords
Examples
a(16)=4 because 4^16-1 = 3*5*17*257*65537, which are the 5 algebraic factors.
Links
- Don Reble, Table of n, a(n) for n = 1..47
- Don Reble, Table of n, a(n) for n = 1..99 (with question marks at the unknown entries)
Programs
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Maple
f:= proc(n) local p, C, x, d; C:= [seq(numtheory:-cyclotomic(d,x), d = numtheory:-divisors(n) minus {1})]; p:= 1; do p:= nextprime(p); if andmap(isprime, subs(x=p+1, C)) then return p+1 fi od: end proc: map(f, [$1..29]); # Robert Israel, Jan 31 2018 -
Mathematica
Table[With[{d = Divisors@ n}, SelectFirst[Range[10^3], AllTrue[Cyclotomic[d, #], PrimeQ] &]], {n, 11}] (* Michael De Vlieger, Jan 31 2018 *) -
PARI
for(d=1,17,ds=divisors(d); print("Searching for d|"d":"ds); forprime(p=2,499999,okc=1; for(c=2,length(ds), if(!isprime(subst(polcyclo(ds[c]),x,p+1)),okc=0; break)); if(okc, for(c=1,length(ds), print("Phi("ds[c]","p+1")="subst(polcyclo(ds[c]),x,p+1))); break))) -
PARI
isok(n, x) = {fordiv(n, d, if (! isprime(polcyclo(d, x)), return(0));); return(1);} a(n) = {my(x=2); while (! isok(n, x), x++); x;} \\ Michel Marcus, Jan 31 2018
Extensions
Corrected and extended by Don Reble, Feb 03 2014
Comments