

A225563


Numbers whose totienttrajectory can be partitioned into two sets with the same sum.


2



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

1,1


COMMENTS

The "totienttrajectory" 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 totienttrajectories 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 totienttrajectory 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

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.


EXAMPLE

17 is in the sequence because its totienttrajectory is {17,16,8,4,2,1}, which can be partitioned into 17+4+2+1 = 16+8.


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 *)


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))))*21


CROSSREFS

Cf. A008683, A003434, A007755, A049108, A002202, A000010, A083207.
Sequence in context: A309325 A192863 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



