OFFSET
0,1
COMMENTS
Though this sequence rarely contains primes (see A232448), most of its members tend to contain a few very large prime factors. The name stems from 'Belphegor's Prime', a(13), which was so named by Clifford Pickover (see link). [Comment corrected by N. J. A. Sloane, Dec 14 2015]
a(-1) = 767 is also an integer and a palindrome (but, being 101+666, lacks digit sequence 666). It is composite with prime factor 13 (as are all a(n) where n is either 3 or 5 mod 6). Note that 13 is the prime number that divides the largest fraction of the a(n) asymptotically, namely 1/3 of them. - Jeppe Stig Nielsen, Dec 15 2025
LINKS
Stanislav Sykora, Table of n, a(n) for n = 0..497
Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)
Clifford A. Pickover, Belphegor's Prime: 1000000000000066600000000000001
Simon Singh, Homer Simpson's scary math problems. BBC News. Retrieved 31 October 2013.
Eric Weisstein's World of Mathematics, Belphegor Number
Wikipedia, Belphegor's prime
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).
FORMULA
a(n) = 666*10^(n+1)+100^(n+2)+1.
G.f.: (16661 - 782770*x + 767000*x^2) / ((1 - x)*(1 - 10*x)*(1 - 100*x)). [Bruno Berselli, Nov 25 2013]
MATHEMATICA
LinearRecurrence[{111, -1110, 1000}, {16661, 1066601, 100666001}, 15] (* Paolo Xausa, Mar 27 2025 *)
PROG
(PARI) Belphegor(k)=(10^(k+3)+666)*10^(k+1)+1; nmax = 498; v = vector(nmax); for (n=0, #v-1, v[n+1]=Belphegor(n))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Stanislav Sykora, Nov 24 2013
STATUS
approved