A309268 Carmichael numbers m such that A309132(m) < m.

561, 1105, 46657, 52633, 188461, 670033, 825265, 838201, 1082809, 2455921, 2628073, 4463641, 4767841, 5632705, 8830801, 11119105, 13187665, 16778881, 18307381, 18900973, 21584305, 22665505, 31146661, 31405501, 31692805, 34657141, 36765901, 38624041, 40280065

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Jul 20 2019

Keywords

• nonn

Comments

A309132(m) divides m for all Carmichael numbers m, but apparently most of them equal A309132(m). Of the first 10000 Carmichael numbers, only 1341 are in this sequence.

The ratios a(n)/A309132(a(n)) are 3, 5, 13, 7, 133, 7, 133, 7, 7, 793, 7, 13, 13, ...

By Jonathan Sondow's theorem (cf. comments in A309132), these are Carmichael numbers m such that denominator(Sum_{prime p|m}1/p - 1/m) < m, i.e., A326690(m) < m.

Problem: are there Carmichael numbers m such that A309132(m) is prime? Equivalently, Carmichael numbers m such that A326690(m) is prime. None exist below 2^64. Conjecture: there are no such Carmichael numbers.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A309132, A326690.

Subsequence of A002997 and A309235.

Programs

Mathematica

aQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]] && Denominator[ Total@(1/FactorInteger[n][[;; , 1]]) - 1/n] < n; Select[Range[10^6], aQ]