This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A182208 #32 Feb 16 2025 08:33:13 %S A182208 1729,2821,6601,8911,15841,41041,52633,63973,101101,126217,172081, %T A182208 188461,670033,748657,825265,838201,997633,1033669,1082809,1773289, %U A182208 2628073,4463641,4909177,6840001,7995169,8719921,8830801,9585541,9890881 %N A182208 Carmichael numbers divisible by 7. %C A182208 Conjecture: Any Carmichael number C divisible by 7 can be written in one of two ways: (1) C=7*(6m+1)*(6n+1), where m and n are natural numbers or (2) C=7*(6m-1)*(6n-1), where m and n are natural numbers. In other words, there aren’t Carmichael numbers divisible by 7 of the form C=7*(6m+1)*(6n-1). Checked for the first 27 Carmichael numbers divisible by 7. Note: a Carmichael number with more than 3 prime divisors can be written (sometimes) in both ways: 41041 = 7*11*13*41 = 7*13*451 (form 1) = 7*11*533 = 7*41*143 (form 2). %C A182208 Observation: in the first 100 Carmichael numbers with three prime divisors (not divisible by 3) there is no one to can be written as (6x+1)(6y+1)(6z-1), they are all of the form (6x+1)(6y+1)(6z+1), (6x-1)(6y-1)(6z-1) or (6x+1)(6y-1)(6z-1). Would not that be enough to make an assumption that there are no such Carmichael numbers with three prime divisors, or even more, that aren't Carmichael numbers even with more than three divisors to can be written this way? %C A182208 The conjecture follows from Korselt's criterion. - _Charles R Greathouse IV_, Oct 02 2012 %H A182208 Charles R Greathouse IV, <a href="/A182208/b182208.txt">Table of n, a(n) for n = 1..10000</a> %H A182208 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a> %t A182208 CarmichaelNbrQ[n_] := ! PrimeQ[n] && Mod[n, CarmichaelLambda@ n] == 1; 7 Select[ Range[2500000], CarmichaelNbrQ[ 7#] &] (* _Robert G. Wilson v_, Aug 24 2012 *) %o A182208 (PARI) Korselt(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,2]>1||(n-1)%(f[i,1]-1),return(0)));1 %o A182208 forstep(n=49,1e6,42,if(Korselt(n),print1(n", "))) \\ _Charles R Greathouse IV_, Oct 02 2012 %Y A182208 Intersection of A002997 (Carmichael) and A008589 (multiples of 7). - _Michel Marcus_, Oct 11 2016 %K A182208 nonn %O A182208 1,1 %A A182208 _Marius Coman_, Apr 18 2012 %E A182208 Corrected by _Robert G. Wilson v_, Aug 24 2012