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

10, 22, 28, 30, 44, 46, 52, 54, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 110, 116, 126, 130, 136, 138, 140, 148, 150, 164, 166, 172, 178, 184, 190, 196, 198, 204, 208, 210, 212, 222, 226, 228, 238, 250, 260, 262, 268, 270, 282, 292, 294, 296, 306, 310, 316, 328, 330

Offset: 1

Author

Torlach Rush, Apr 29 2018

Keywords

• nonn

Comments

Terms of this sequence are totients selected by prime replicators of totients not terms of this sequence.

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.

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.

Examples

			10 is a term because the largest prime factor of 11 and 22, the solutions of phi(10) is 11.
2 is not a term because there is no common largest prime factor of 3, 4 and 6, the solutions of phi(2).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• K. B. Stolarski and S. Greenbaum, A Ratio Associated with phi(x) = n, The Fibonacci Quarterly, Volume 23, Number 3, August 1985, pp. 265-269.

Crossrefs

Cf. A000010, A002202, A085713.

Subsequence of A303745.

Programs

Maple

filter:= proc(n) local L,q;
  L:= numtheory:-invphi(n);
  if nops(L) = 0 then return false fi;
  q:= max(numtheory:-factorset(L[1]));
  andmap(t -> max(numtheory:-factorset(t))=q, L[2..-1]);
end proc:
select(filter, [seq(i,i=2..1000,2)]); # Robert Israel, Jun 25 2018

PARI

isok(n) = if (n > 1, #Set(apply(x->vecmax(factor(x)[,1]), invphi(n))) == 1); \\ Michel Marcus, May 13 2018