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

1,1

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Max Alekseyev, PARI scripts for various 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.

Example

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).

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

Prog

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

Crossrefs

Cf. A000010, A002202, A085713.

Subsequence of A303745.

Sequence in context: A063555 A228010 A303745 * A303747 A007366 A302280

Adjacent sequences: A303743 A303744 A303745 * A303747 A303748 A303749

Keyword

nonn

Author

Torlach Rush, Apr 29 2018

Status

approved