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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 (list; graph; refs; listen; history; text; internal format)
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
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(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
Intersection of A303745 and A303746.
Sequence in context: A228010 A303745 A303746 * A007366 A302280 A350627
KEYWORD
nonn
AUTHOR
Torlach Rush, Apr 29 2018
EXTENSIONS
Definition clarified by Robert Israel, Jun 25 2018
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)