A006972 Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.

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

Offset: 1

Author

Richard Pinch and Jeffrey Shallit

Keywords

• nonn

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.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

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.

Programs

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} & ]

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