OFFSET
1,1
COMMENTS
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
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
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).
Sridhar Tamilvanan and Subramani Muthukrishnan, On Lucas-Carmichael Integer, arXiv:2311.08012 [math.NT], 2023.
Daniel Suteu, Table of n, a(n) for terms a(n) < 10^15.
Samuel S. Wagstaff, Jr., Ramanujan's Taxicab Number and its Ilk, Purdue Univ. (2024). See p. 2.
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.
MAPLE
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
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
(PARI)
lucas_carmichael(A, B, k) = A=max(A, vecprod(primes(k+1))\2); (f(m, l, lo, k) = my(list=List()); my(hi=min(sqrtint(B+1)-1, sqrtnint(B\m, k))); if(lo > hi, return(list)); if(k==1, lo=max(lo, ceil(A/m)); my(t=lift(-1/Mod(m, l))); while(t < lo, t += l); forstep(p=t, hi, l, if(isprime(p), my(n=m*p); if((n+1)%(p+1) == 0, listput(list, n)))), forprime(p=lo, hi, if(gcd(m, p+1) == 1, list=concat(list, f(m*p, lcm(l, p+1), p+1, k-1))))); list); f(1, 1, 3, k);
upto(n) = my(list=List()); for(k=3, oo, if(vecprod(primes(k+1))\2 > n, break); list=concat(list, lucas_carmichael(1, n, k))); vecsort(Vec(list)); \\ Daniel Suteu, Dec 01 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved