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A167766
Minimum numbers whose phi of phi are multiples of the n-th prime: the n-th term is the minimum integer x such that: prime(n) | phi(phi(x)), prime(n) being the n-th prime.
3
5, 19, 23, 59, 47, 107, 479, 383, 283, 467, 1367, 1187, 167, 347, 1319, 643, 2837, 2203, 2153, 3413, 587, 5693, 1997, 359, 5827, 1619, 2063, 2999, 4799, 3167, 1019, 1579, 5483, 3343, 7159, 3023, 12569, 1307, 4679, 2083, 719, 3623, 4597, 3863, 18917, 4783
OFFSET
1,1
COMMENTS
These minimal integers are always prime. To be clear, the phi function referred to here is Euler's totient function.
LINKS
Eric Weisstein's World of Mathematics, Euler's Totient Function
EXAMPLE
The first term is 5. phi(5) = 4 and phi(4)=2. 2 is a multiple of the first prime 2. 5 is the lowest such number x where 2 divides phi(phi(x)).
MAPLE
with(numtheory): P:=proc(n) local a, k; a:=ithprime(n);
for k from 1 to 10^3 do if frac(phi(phi(ithprime(k)))/a)=0
then RETURN(ithprime(k)); break; fi; od; end:
seq(P(i), i=1..46); # Paolo P. Lava, Oct 10 2018
MATHEMATICA
a[n_] := (p=Prime[n]; k=1; While[k++; x=Prime[k]; Mod[ EulerPhi[ EulerPhi[x]], p] != 0]; x); Table[a[n], {n, 50}] (* Jean-François Alcover, Sep 14 2011 *)
PROG
(PARI) /* not the most efficient implementation */ ppp(a, b)= { forprime(p=a, b, v = 2*p + 1; v2 = 1; minv = 100000000; while (v2 <= minv || v <=minv, /* print ("Checking ", v, " for ", p); */ while(!isprime(v), v += 2*p /*; print ("Checking ", v, " for ", p)*/ ); if (v%(p*p)==1, /* don't do this step if: p^2 | v-1 */ v2 = v , v2 = 2*v + 1; while (!isprime(v2) && v2 < minv, v2 += 2*v ) ); if (v2 < minv, minv = v2; ); v += 2*p ); print (p, " => ", minv) ) }
CROSSREFS
Cf. A010554.
Sequence in context: A191084 A146509 A062340 * A106957 A236167 A022143
KEYWORD
easy,nice,nonn
AUTHOR
Fred Schneider, Nov 11 2009
STATUS
approved