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A006972
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Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.
(Formerly M5450)
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48
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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
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1,1
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COMMENTS
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Wright proves that this sequence is infinite (Main Theorem 2). - Charles R Greathouse IV, Nov 03 2015
Conjecture: if k = p*q*r, p = a*d - 1, q = b*d - 1, r = c*d - 1 are distinct odd primes, with d = gcd(p + 1, q + 1, r + 1) and a*b*c*d divides k + 1, then k is a Lucas-Carmichael number. - Davide Rotondo, Dec 23 2020
A composite k is a Lucas-Carmichael number if and only if k | A322702(k+1). - Thomas Ordowski, May 06 2021
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REFERENCES
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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).
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LINKS
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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 video (2015)
Wikipedia, Lucas-Carmichael number
Thomas Wright, There are infinitely many elliptic Carmichael numbers
Thomas Wright, There are infinitely many elliptic Carmichael numbers, arXiv:1609.00231 [math.NT], 2016.
Index entries for sequences related to Carmichael numbers.
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MAPLE
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with(numtheory):
a:= proc(n) option remember; local k; for k from 1+
`if`(n=1, 3, a(n-1)) while isprime(k) or not issqrfree(k)
or add(irem(k+1, i+1), i=factorset(k))>0 do od; k
end:
seq(a(n), n=1..15); # Alois P. Heinz, Apr 05 2018
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MATHEMATICA
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Select[ Range[ 2, 10^6 ], !PrimeQ[ # ] && Union[ Transpose[ FactorInteger[ # ] ][ [ 2 ] ] ] == {1} && Union[ Mod[ # + 1, Transpose[ FactorInteger[ # ] ][ [ 1 ] ] + 1 ] ] == {0} & ]
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PROG
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(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
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CROSSREFS
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Intersection of A024556 and A056729.
Cf. A216925, A216926, A216927, A217002, A217003, A217091 (terms having 3, 4, 5, 6, 7 and 8 factors).
Cf. A216929, A322702.
Sequence in context: A158317 A227008 A253597 * A216925 A292573 A299213
Adjacent sequences: A006969 A006970 A006971 * A006973 A006974 A006975
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KEYWORD
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nonn
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AUTHOR
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Richard Pinch and Jeffrey Shallit
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STATUS
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approved
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