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A303747
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Totients t for which gcd({x: phi(x)=t}) equals the largest prime factor of each member of {x: phi(x)=t}.
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2
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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
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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