OFFSET
1,1
COMMENTS
From Michael De Vlieger, May 21 2017: (Start)
Conjecture: all terms are even.
Let b(n) = differences of a(n). Many of the terms in b(n) are in A060735(n), i.e., b(n) >= A002110(n) and divisible by A002110(n). The smallest b(n) that is not in A060735 is b(450) = 66; b(450) > A002110(n) but not divisible by it; instead it is divisible by A002110(n-1). b(n) seems to "prefer" values of A002110(n) as n increases. There is a run of 22 values of 2 starting at b(2), of 22 values of 6 starting at b(178), of 7 values of 210 starting at b(543), and 3 values of 2310 starting at b(577). In the case of A002110(n) with n < 3, the length of runs of A002110(n) is greater than that of runs of any other number of comparable magnitude. The last observed position of A002110(n) in b(n) is {201, 442, (557), ...}.
Questions: are there increasingly more terms such that b(n) is not in A060735 as n increases? Are terms such that b(n) is in A060735 prevalent? Why does b(n) seem to "prefer" values in A002110 at certain magnitudes of n as n increases? Are there differences of a(n) = A002110(k) at positions greater than those observed, and if not, why? (End)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..586 (a(n) < 10^4).
Michael De Vlieger, Values of a(n), their differences, and multiplicity notation of their differences.
Wikipedia, Reduced residue system
EXAMPLE
n=30 is the largest extremal example whose reduced residue system consists only of primes and 1 (see A048597); n=8 Phi(8)=4, reduced residue system (8)={1,3,5,7} n=32 Phi(32)=16, reduced residue system (32)={1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31} of which only {1,9,15,21,25,27} are not primes,10 are primes: 10>6 thus 32 belongs here.
MATHEMATICA
Select[Range@ 170, Function[n, Count[Range@ n, _?(PrimeQ@ # && CoprimeQ[n, #] &)] > EulerPhi[n]/2]] (* Michael De Vlieger, May 21 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved