A213812 a(n) = smallest m for which the n-th Carmichael number A002997(n) can be written as p^2*(m+1) - p*m.

1, 3, 4, 2, 2, 3, 1, 1, 2, 7, 24, 4, 4, 7, 47, 80, 9, 1, 23, 2, 46, 15, 24, 21, 24, 1, 1, 76, 8, 21, 16, 14, 6, 2, 150, 16, 8, 16, 3, 156, 36, 232, 2, 13, 10, 788, 40, 25, 2, 4, 123, 12, 44, 16, 8, 207, 226, 462, 92, 6

Offset: 1

Author

Marius Coman, Jun 20 2012

Keywords

• nonn

Comments

The corresponding values of p are (we write the Carmichael number in brackets): 17(561), 17(1105), 19(1729), 29(2465), 31(2821), 41(6601), 67(8911), 73(10585), 73(15841), 61(29341), 41(41041), 97(46657), 103(52633), 89(62745), 37(63973), 31(75361), 101(101101), 241(115921), 73(126217), 233(162401), 61(172081), 109(188461), 101(252601), 113(278545), 109(294409), 397(314821), 409(334153), 67(340561), 211(399001), 137(410041), 163(449065), 181(488881), 271(512461), 421(530881), 61(552721), 197(656601), 271(658801), 199(670033), 433(748657), 73(825265), 151(838201), 61(852841), 577(997633), 271(1024651), 307(1033669), 37(1050985), 163(1082809), 211(1152271), 631(1193221), 541(1461241), 113(1569457), 353(1615681), 199(1773289), 331(1857241), 461(1909001), 101(2100901), 97(2113921), 73(2433601), 163(2455921), 599(2508013).

Any Carmichael number C can be written as C = p^2*(n+1) - p*n, where p is any prime divisor of C (it can be seen that the smallest n is obtained for the biggest prime divisor).

The formula C = p^2*(n+1) - p*n is equivalent to C = p^2*m - p*(m-1) = p^2*m - p*m + p, equivalent to p^2 - p divides C - p, which is a direct consequence of Korselt’s criterion.

It can be shown from p - 1 divides C - 1 not that just p^2 - p divides C - p but even that p^2 - p divides C - p^k (if C > p^k) or p^k - C (if p^k > C) which leads to the generic formula for a Carmichael number: C = p^k + n*p^2 - n*p (if C > p^k) or C = p^k - n*p^2 + n*p (if p^k > C) for any p prime divisor of C and any k natural number.

The formulas generated giving values of k seems to be very useful in the study of Fermat pseudoprimes; also, the composite numbers C for which the equation C = p^k - n*p^2 + n*p gives, over the integers, as solutions, all their prime divisors, for a certain k, deserve further study.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Carmichael Number

Crossrefs

Cf. A002997.

Programs

PARI

Car=[561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217]; \\ use more terms of A002997 as desired
apply(C->my(f=factor(C)[,1],p=f[#f],p2=p^2); (C-p2)/(p2-p), Car) \\ Charles R Greathouse IV, Jul 05 2017