

A006972


LucasCarmichael numbers: squarefree composite numbers n such that p  n => p+1  n+1.
(Formerly M5450)


21



399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535, 76751, 80189, 81719, 88559, 90287, 104663, 117215, 120581, 147455, 152279, 155819, 162687, 191807, 194327, 196559, 214199
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OFFSET

1,1


COMMENTS

Number of terms less than 10^n: 0, 0, 2, 8, 26, 60, 135, 323, 791, 1841, 4216, 9967, ..., . (A216929)  Robert G. Wilson v, Feb 11 2015
In the first 10000 terms 430 are congruent to 1 (mod 10), 415 are == 3 (mod 10), 892 are == 5 (mod 10), 395 are == 7 (mod 10) and 7868 are == 9 (mod 10).  Robert G. Wilson v, Feb 11 2015
Wright proves that this sequence is infinite (Main Theorem 2).  Charles R Greathouse IV, Nov 03 2015


REFERENCES

J.M. De Koninck, Ces nombres qui nous fascinent, Entry 399, p. 89, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Paolo P. Lava and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 550 terms from Paolo P. Lava)
Ed Copeland and Brady Haran, Something special about 399  Numberphile (2015)
Wikipedia, LucasCarmichael number
Thomas Wright, There are infinitely many elliptic Carmichael numbers
Thomas Wright, There are infinitely many elliptic Carmichael numbers, arXiv:1609.00231 [math.NT], (September 2016)
Index entries for sequences related to Carmichael numbers.


MATHEMATICA

Select[ Range[ 2, 10^6 ], !PrimeQ[ # ] && Union[ Transpose[ FactorInteger[ # ] ][ [ 2 ] ] ] == {1} && Union[ Mod[ # + 1, Transpose[ FactorInteger[ # ] ][ [ 1 ] ] + 1 ] ] == {0} & ]


PROG

(PARI) is(n)=my(f=factor(n)); for(i=1, #f[, 1], if((n+1)%(f[i, 1]+1)  f[i, 2]>1, return(0))); #f[, 1]>1 \\ Charles R Greathouse IV, Sep 23 2012


CROSSREFS

Intersection of A024556 and A056729.
Cf. A216925, A216926, A216927, A217002, A217003, A217091 (terms having 3, 4, 5, 6, 7 and 8 factors).
Sequence in context: A158317 A227008 A253597 * A216925 A206536 A065767
Adjacent sequences: A006969 A006970 A006971 * A006973 A006974 A006975


KEYWORD

nonn


AUTHOR

Richard Pinch and Jeffrey Shallit


STATUS

approved



