A212843 Carmichael numbers that have only prime divisors of the form 10k+1.

252601, 399001, 512461, 852841, 1193221, 1857241, 1909001, 2100901, 3828001, 5049001, 5148001, 5481451, 6189121, 7519441, 8341201, 9439201, 10024561, 10837321, 14676481, 15247621, 17236801, 27062101, 29111881, 31405501, 33302401, 34657141, 40430401, 42490801

Offset: 1

Author

Marius Coman, May 28 2012

Keywords

• nonn

Comments

Conjecture: only Carmichael numbers of the form 10n+1 can have prime divisors of the form 10k+1 (but not all Carmichael numbers of the form 10n+1 have prime divisors of the form 10k+1).

Checked up to Carmichael number 4954039956700380001.

Conjecture: all Carmichael numbers C (not only with three prime divisors) of the form 10n+1 that have only prime divisors of the form 10k+1 can be written as C = (30a+1)*(30b+1)*(30c+1), C = (30a+11)*(30b+11)*(30c+11), or C = (30a+1)*(30b+11)*(30c+11). In other words, there are no numbers of the form C = (30a+1)*(30b+1)*(30c+11).

Checked for all Carmichael numbers from the sequence above.

The first conjecture is a consequence of Korselt's criterion. - Charles R Greathouse IV, Oct 02 2012

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• E. W. Weisstein, Carmichael Number

Crossrefs

Subsequence of A004615.

Extensions

Terms corrected by Charles R Greathouse IV, Oct 02 2012