A232449 The palindromic Belphegor numbers: (10^(n+3)+666)*10^(n+1)+1.

16661, 1066601, 100666001, 10006660001, 1000066600001, 100000666000001, 10000006660000001, 1000000066600000001, 100000000666000000001, 10000000006660000000001, 1000000000066600000000001, 100000000000666000000000001, 10000000000006660000000000001, 1000000000000066600000000000001

Offset: 0

Author

Stanislav Sykora, Nov 24 2013

Keywords

• nonn
• easy

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]

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).

Crossrefs

Cf. A232448, A232450, A232451.

Subsequence of A118598.

Programs

Mathematica

A232449[n_] := 100^(n+2) + 666*10^(n+1) + 1; Array[A232449, 15, 0] (* or *)
LinearRecurrence[{111, -1110, 1000}, {16661, 1066601, 100666001}, 15] (* Paolo Xausa, Mar 27 2025 *)

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))

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]