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A225563 Numbers whose totient-trajectory can be partitioned into two sets with the same sum. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000

Christian N. K. Anderson, Decomposition of the first 1000 terms.

EXAMPLE

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.

PROG

(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

CROSSREFS

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

Sequence in context: A248608 A309325 A143450 * A294923 A005842 A204458

Adjacent sequences:  A225560 A225561 A225562 * A225564 A225565 A225566

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)