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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

%I

%S 5,19,23,59,47,107,479,383,283,467,1367,1187,167,347,1319,643,2837,

%T 2203,2153,3413,587,5693,1997,359,5827,1619,2063,2999,4799,3167,1019,

%U 1579,5483,3343,7159,3023,12569,1307,4679,2083,719,3623,4597,3863,18917,4783

%N 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.

%C These minimal integers are always prime. To be clear, the phi function referred to here is Euler's totient function.

%H Donovan Johnson, <a href="/A167766/b167766.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Euler's Totient Function</a>

%e 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)).

%p with(numtheory): P:=proc(n) local a,k; a:=ithprime(n);

%p for k from 1 to 10^3 do if frac(phi(phi(ithprime(k)))/a)=0

%p then RETURN(ithprime(k)); break; fi; od; end:

%p seq(P(i),i=1..46); # _Paolo P. Lava_, Oct 10 2018

%t 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 *)

%o (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) ) }

%Y Cf. A010554.

%K easy,nice,nonn

%O 1,1

%A _Fred Schneider_, Nov 11 2009

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Last modified October 21 03:24 EDT 2019. Contains 328291 sequences. (Running on oeis4.)