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A303747 Totients t for which gcd({x: phi(x)=t}) equals the largest prime factor of each member of {x: phi(x)=t}.

%I A303747 #51 Jan 07 2025 10:04:10
%S A303747 10,22,28,30,44,46,52,56,58,66,70,78,82,92,102,104,106,116,126,130,
%T A303747 136,138,140,148,150,164,166,172,178,184,190,196,198,204,208,210,212,
%U A303747 222,226,228,238,250,260,262,268,270,282,292,296,306,310,316,328,330,332,344,346
%N A303747 Totients t for which gcd({x: phi(x)=t}) equals the largest prime factor of each member of {x: phi(x)=t}.
%C A303747 Terms of this sequence are totients selected by prime replicators of totients not terms of this sequence.
%C A303747 Following are some examples of terms and their corresponding prime replicators for increasing cardinality of solutions:
%C A303747 #({x: phi(x)=t}) = 2: {(10,11),(22,23),(28,29),(30,31),(46,47),(52,53),...}
%C A303747 #({x: phi(x)=t}) = 3: {(44,23),(56,29),(92,47),(104,53),(116,59),(140,71),...}
%C A303747 #({x: phi(x)=t}) = 4: {(184,47),(208,53),(328,83),(424,107),(664,167),...}
%C A303747 #({x: phi(x)=t}) = 5: {(368,47),(416,53),(656,83),(848,107),(1328,167),...}
%C A303747 #({x: phi(x)=t}) = 6: {(984,83),(1272,107),(6024,503),(7824,653),...}
%C A303747 ...
%C A303747 Denote the starting or seed totient for each of the above TS and we have {1,2,4,8,12,...}. We then have a relation between all of the terms (T) and their corresponding primes (P), which is T = (P * TS) - TS.
%C A303747 The values of the GCD of the solutions of terms of this sequence are the terms of A058340.
%H A303747 Robert Israel, <a href="/A303747/b303747.txt">Table of n, a(n) for n = 1..10000</a>
%H A303747 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP scripts for miscellaneous problems</a>
%H A303747 K. B. Stolarski and S. Greenbaum, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/23-3/stolarsky.pdf">A Ratio Associated with phi(x) = n</a>, The Fibonacci Quarterly, Volume 23, Number 3, August 1985, pp. 265-269.
%e A303747 10 is a term because the largest prime factor of 11 and 22, the solutions of phi(x)=10 is 11 which is also the greatest common divisor of the solutions of phi(x)=10.
%e A303747 54 is not a term because while 3 is the largest prime factor of solutions phi(x)=54, 3 <> gcd({x: phi(x)=54}) = 81.
%p A303747 filter:= proc(n) local L,q;
%p A303747   L:= numtheory:-invphi(n);
%p A303747   if nops(L) = 0 then return false fi;
%p A303747   q:= igcd(op(L));
%p A303747   if not isprime(q) then return false fi;
%p A303747   andmap(t -> max(numtheory:-factorset(t))=q, L);
%p A303747 end proc:
%p A303747 select(filter, [seq(i,i=2..1000,2)]); # _Robert Israel_, Jun 25 2018
%o A303747 (PARI) isok(n) = my(v=invphi(n)); ((g=gcd(v)) > 1) && (s = Set(apply(x->vecmax(factor(x)[,1]), invphi(n)))) && (#s == 1) && (s[1] == g); \\ _Michel Marcus_, May 13 2018
%Y A303747 Cf. A000010, A002202, A058340, A085713.
%Y A303747 Intersection of A303745 and A303746.
%K A303747 nonn
%O A303747 1,1
%A A303747 _Torlach Rush_, Apr 29 2018
%E A303747 Definition clarified by _Robert Israel_, Jun 25 2018