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A225563
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Numbers whose totient-trajectory can be partitioned into two sets with the same sum.
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2
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3, 5, 7, 9, 11, 13, 15, 17, 25, 27, 31, 33, 35, 39, 41, 49, 51, 55, 61, 65, 69, 77, 81, 85, 87, 91, 95, 97, 103, 111, 115, 119, 121, 123, 125, 133, 137, 141, 143, 145, 153, 155, 159, 161, 175, 183, 185, 187, 193, 201, 203, 205, 209, 213, 215, 217, 219, 221
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OFFSET
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1,1
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COMMENTS
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The "totient-trajectory" of a number m is the sequence obtained by starting with m and repeatedly applying the map x -> phi(x) (cf. A000010) until reaching 1.
Because all totient-trajectories contain only even numbers apart from the final 1 and (perhaps) the initial term ending in 1, only odd numbers will be in the sequence.
Conjecture: No totient-trajectory can be partitioned into an odd number of sets with the same sum.
Observation: for the first 1000 terms, numbers ending in 5 are more than twice as frequent as those ending in any other number.
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LINKS
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EXAMPLE
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17 is in the sequence because its totient-trajectory is {17,16,8,4,2,1}, which can be partitioned into 17+4+2+1 = 16+8.
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MATHEMATICA
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totQ[n_] := Module[{it = Most@FixedPointList[EulerPhi, n], sum, x}, sum = Plus @@ it; If[OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, it}], x][[1 +sum/2]] > 0]]; Select[Range[221], totQ] (* Amiram Eldar, May 24 2020 *)
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PROG
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(R)library(numbers); totseq<-function(x) { while(x[length(x)]>1) x[length(x)+1]=eulersPhi(x[length(x)]); x };
eqsum<-function(xvec) {
mkgrp<-function(grp) {
if(length(grp)==length(xvec)) {
tapply(xvec, grp, sum)->tot;
if(length(tot)==2) if(tot[1]==tot[2]) {faxp<<-grp; return(T)}; return(F);
}
ifelse(mkgrp(c(grp, 1)), T, mkgrp(c(grp, 2)));
}
ifelse(length(xvec)<2, F, mkgrp(c()));
}
which(sapply(2*(1:100)-1, function(x) eqsum(totseq(x))))*2-1
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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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