

A303747


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


2



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
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OFFSET

1,1


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.


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. 265269.


EXAMPLE

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.


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


PROG

(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


CROSSREFS

Cf. A000010, A002202, A058340, A085713.
Intersection of A303745 and A303746.
Sequence in context: A228010 A303745 A303746 * A007366 A302280 A350627
Adjacent sequences: A303744 A303745 A303746 * A303748 A303749 A303750


KEYWORD

nonn


AUTHOR

Torlach Rush, Apr 29 2018


EXTENSIONS

Definition clarified by Robert Israel, Jun 25 2018


STATUS

approved



