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

10, 22, 28, 30, 44, 46, 52, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 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, 296, 306, 310, 316, 328, 330, 332, 344, 346

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.

Following are some examples of terms and their corresponding prime replicators for increasing cardinality of solutions:

#({x: phi(x)=t}) = 2: {(10,11),(22,23),(28,29),(30,31),(46,47),(52,53),...}

#({x: phi(x)=t}) = 3: {(44,23),(56,29),(92,47),(104,53),(116,59),(140,71),...}

#({x: phi(x)=t}) = 4: {(184,47),(208,53),(328,83),(424,107),(664,167),...}

#({x: phi(x)=t}) = 5: {(368,47),(416,53),(656,83),(848,107),(1328,167),...}

#({x: phi(x)=t}) = 6: {(984,83),(1272,107),(6024,503),(7824,653),...}

...

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.

The values of the GCD of the solutions of terms of this sequence are the terms of A058340.

Examples

			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.
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP scripts for miscellaneous problems
• 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, A058340, A085713.

Intersection of A303745 and A303746.

Programs

Maple

filter:= proc(n) local L,q;
  L:= numtheory:-invphi(n);
  if nops(L) = 0 then return false fi;
  q:= igcd(op(L));
  if not isprime(q) then return false fi;
  andmap(t -> max(numtheory:-factorset(t))=q, L);
end proc:
select(filter, [seq(i,i=2..1000,2)]); # Robert Israel, Jun 25 2018

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

Extensions

Definition clarified by Robert Israel, Jun 25 2018