Carol number

Given n, compute 4n-2n+1-1 or (2n-1)2-2 or (22n-1)-2n+1 or


Any of these formulas gives the Carol numberMathworldPlanetmath for n. Regardless of how they’re computed, these numbers are almost repunitsMathworldPlanetmath in binary, needing only the addition of 2n+1 to become so.

The first few Carol numbers are 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, 4190207, 16769023 (listed in A093112 of Sloane’s OEIS). Every third Carol number is divisible by 7, thus prime Carol numbers can’t have n=3x+2 (except of course for n=2. The largest Carol number known to be prime is (2248949-1)2-2, found by Japke Rosink using MultiSieve and OpenPFGW in March 2006.

Title Carol number
Canonical name CarolNumber
Date of creation 2013-03-22 16:13:10
Last modified on 2013-03-22 16:13:10
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11N05