A225563 Numbers whose totient-trajectory can be partitioned into two sets with the same sum.

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

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 10 2013

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Christian N. K. Anderson)
• Christian N. K. Anderson, Decomposition of the first 1000 terms.

Crossrefs

Cf. A008683, A003434, A007755, A049108, A002202, A000010, A083207.

Programs

Mathematica

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 *)

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

Extensions

Edited by N. J. A. Sloane, May 17 2013