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k-Lehmer numbers
A number    is a  -Lehmer number if    divides  .

Since    when    is prime, all prime numbers are  -Lehmer numbers.

Every number which is  -Lehmer is also  -Lehmer, and thus for simplicity I will call a number  -Lehmer only if it is not  -Lehmer, and I will consider only composite Lehmer numbers.

The existence of a composite 1-Lehmer number (usually simply called Lehmer number) is still an open problem and several results have been proved about these numbers (which probably do not exist). For example, Cohen and Hagis have proved that such a number, if it exists, must be greater than    and be the product of at least 14 primes.

The following table reports the smallest  -Lehmer number for    from 2 to 36.

 2 561 9 771 16 494211 23 16711935 30 8053383171 3 15 10 43435 17 196611 24 126027651 31 4294967295 4 451 11 3855 18 2089011 25 50529027 32 32212942851 5 51 12 31611 19 983055 26 756493591 33 90665917447 6 679 13 13107 20 8061051 27 252645135 34 129352336131 7 255 14 272163 21 3342387 28 4446487299 35 362186539779 8 2091 15 65535 22 31580931 29 858993459 36 972094264435

Grau & Antonio M. Oller-Marcén have proved several results. For example, that every Carmichael number is also a  -Lehmer number.

The first  -Lehmer numbers are 15, 51, 85, 91, 133, 247, 255, 259, 435, 451, 481, 511, 561, 595, 679, 703, 763, 771, 949, 1105, 1111, 1141, 1261, 1285, 1351, 1387, 1417, 1615, 1695, 1729 more terms

You can download a zipped text file (kLehmer_up_1e12.zip) (length = 9.3 MB), containing the 2103055  -Lehmer numbers up to  .

K-Lehmer numbers can also be... (you may click on names or numbers and on + to get more values)