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%I #41 Nov 24 2024 11:14:59
%S 10,22,28,30,44,46,52,54,56,58,66,70,78,82,92,102,104,106,110,116,126,
%T 130,136,138,140,148,150,164,166,172,178,184,190,196,198,204,208,210,
%U 212,222,226,228,238,250,260,262,268,270,282,292,294,296,306,310,316,328,330
%N Totients t for which {x: phi(x)=t} share the same largest prime factor.
%C Terms of this sequence are totients selected by prime replicators of totients not terms of this sequence.
%C 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 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 Robert Israel, <a href="/A303746/b303746.txt">Table of n, a(n) for n = 1..10000</a>
%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H K. B. Stolarski and S. Greenbaum, <a href="http://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 10 is a term because the largest prime factor of 11 and 22, the solutions of phi(10) is 11.
%e 2 is not a term because there is no common largest prime factor of 3, 4 and 6, the solutions of phi(2).
%p filter:= proc(n) local L,q;
%p L:= numtheory:-invphi(n);
%p if nops(L) = 0 then return false fi;
%p q:= max(numtheory:-factorset(L[1]));
%p andmap(t -> max(numtheory:-factorset(t))=q, L[2..-1]);
%p end proc:
%p select(filter, [seq(i,i=2..1000,2)]); # _Robert Israel_, Jun 25 2018
%o (PARI) isok(n) = if (n > 1, #Set(apply(x->vecmax(factor(x)[,1]), invphi(n))) == 1); \\ _Michel Marcus_, May 13 2018
%Y Cf. A000010, A002202, A085713.
%Y Subsequence of A303745.
%K nonn
%O 1,1
%A _Torlach Rush_, Apr 29 2018