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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==tot) {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.)