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A303746 Totients t for which {x: phi(x)=t} share the same largest prime factor.

%I A303746 #43 Jan 05 2025 19:51:41
%S A303746 10,22,28,30,44,46,52,54,56,58,66,70,78,82,92,102,104,106,110,116,126,
%T A303746 130,136,138,140,148,150,164,166,172,178,184,190,196,198,204,208,210,
%U A303746 212,222,226,228,238,250,260,262,268,270,282,292,294,296,306,310,316,328,330
%N A303746 Totients t for which {x: phi(x)=t} share the same largest prime factor.
%C A303746 Terms of this sequence are totients selected by prime replicators of totients not terms of this sequence.
%C A303746 A303747 a restriction of this sequence gives a relation T = (P * TS) - TS where T is a term, P is the corresponding prime replicator and TS is the starting or seed totient. The relation fails for a(202) = 1210. 1210 does not equal (11 * a(19)) - a(19), i.e., (11 * 110) - 110.
%C A303746 For known terms, the greatest common divisor of the solutions of a(n) is either a power of the largest prime factor of solutions of a(n), or is evenly divisible by same.
%H A303746 Robert Israel, <a href="/A303746/b303746.txt">Table of n, a(n) for n = 1..10000</a>
%H A303746 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A303746 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 A303746 10 is a term because the largest prime factor of 11 and 22, the solutions of phi(10) is 11.
%e A303746 2 is not a term because there is no common largest prime factor of 3, 4 and 6, the solutions of phi(2).
%p A303746 filter:= proc(n) local L,q;
%p A303746   L:= numtheory:-invphi(n);
%p A303746   if nops(L) = 0 then return false fi;
%p A303746   q:= max(numtheory:-factorset(L[1]));
%p A303746   andmap(t -> max(numtheory:-factorset(t))=q, L[2..-1]);
%p A303746 end proc:
%p A303746 select(filter, [seq(i,i=2..1000,2)]); # _Robert Israel_, Jun 25 2018
%o A303746 (PARI) isok(n) = if (n > 1, #Set(apply(x->vecmax(factor(x)[,1]), invphi(n))) == 1); \\ _Michel Marcus_, May 13 2018
%Y A303746 Cf. A000010, A002202, A085713.
%Y A303746 Subsequence of A303745.
%K A303746 nonn
%O A303746 1,1
%A A303746 _Torlach Rush_, Apr 29 2018