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A182151 Carmichael numbers divisible by 31. 1
2821, 15841, 75361, 172081, 399001, 512461, 852841, 1193221, 1857241, 2100901, 2113921, 3146221, 4903921, 5049001, 5481451, 6049681, 8341201, 8927101, 9585541, 10606681, 10837321, 11205601, 18162001, 27062101, 27402481 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: Any Carmichael number C divisible by 31 can be written in one of the following three forms: C = 2790n+31; C = 2790n+31^2 or C = 2790n+31*61, where n is natural.

Examples:

Carmichael numbers of the first form: 2821, 75361, 399001, 2100901.

Carmichael numbers of the second form: 6049681, 10837321, 11205601.

Carmichael numbers of the third form: 15841, 172081, 512461, 852841, 1193221, 1857241, 2113921, 3146221, 4903921, 5049001, 5481451, 8341201, 8927101, 9585541, 10606681, 18162001.

Checked for the first 23 Carmichael numbers divisible by 31.

It follows from Korselt's criterion that such numbers are 31 mod 930, the union of the three residue classes. Thus the conjecture is true. - Charles R Greathouse IV, Oct 02 2012

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

E. W. Weisstein, MathWorld: Carmichael Number

MATHEMATICA

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; 31 Select[ Range@ 1000000, CarmichaelNbrQ[ 31#] &] (* Robert G. Wilson v, Aug 24 2012 *)

PROG

(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

forstep(n=961, 1e5, 930, if(Korselt(n), print1(n", "))) \\ Charles R Greathouse IV, Oct 02 2012

CROSSREFS

Sequence in context: A179159 A248985 A236794 * A182085 A271580 A237063

Adjacent sequences:  A182148 A182149 A182150 * A182152 A182153 A182154

KEYWORD

nonn

AUTHOR

Marius Coman, Apr 18 2012

STATUS

approved

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Last modified October 22 22:22 EDT 2020. Contains 337962 sequences. (Running on oeis4.)