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A020342 Vampire numbers: (definition 1): n has a nontrivial factorization using n's digits.

%I A020342 #72 Mar 14 2025 20:03:56
%S A020342 126,153,688,1206,1255,1260,1395,1435,1503,1530,1827,2187,3159,3784,
%T A020342 6880,10251,10255,10426,10521,10525,10575,11259,11439,11844,11848,
%U A020342 12006,12060,12384,12505,12546,12550,12595,12600,12762,12768,12798,12843,12955,12964
%N A020342 Vampire numbers: (definition 1): n has a nontrivial factorization using n's digits.
%C A020342 Nontrivial means that there must be at least two factors.
%C A020342 For any a(n), 10*a(n) is also in the sequence, and also in A144563. - _M. F. Hasler_, Nov 01 2021
%D A020342 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
%H A020342 Ely Golden, <a href="/A020342/b020342.txt">Table of n, a(n) for n = 1..1000</a> (corrected by _Sean A. Irvine_)
%H A020342 Ely Golden, <a href="/A020342/a020342.py.txt">Sympy program for generating vampire numbers (definition 1)</a>
%H A020342 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_20">Cab and Vampire Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 20, 499-512.
%H A020342 Gordon Hamilton, <a href="http://youtu.be/7efCz2FvUDI">Three integer sequences from recreational mathematics</a>, Video (2013).
%H A020342 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a020/A020342.java">Java program</a> (github)
%e A020342 1395 = 31*9*5, so 1395 is a term.
%e A020342 179739 = 7 * 9 * 9 * 317 so 179739 is a term. - _Sean A. Irvine_, Feb 28 2023
%o A020342 (PARI) is_A020342(n, m=0, D=vecsort(digits(n)))={ if(m && n >= m && vecsort(digits(n))==D, 1, #D<3, m && (D[1]>=m && vecprod(D)==n), n >= m^2, my(S=Set(D), i, C); fordiv(n,f, f < m && next; f*f > n && break; setminus(Set(digits(f)),S) && next; C=D; foreach(digits(f),d, if(i=vecsearch(C,d), C=C[^i], next(2))); C && is_A020342(n\f, f, C) && return(1)))} \\ See A144563 for a function counting the multiplicity of the representation. - _M. F. Hasler_, Nov 01 2021
%Y A020342 Closely related: A014575, A080718, A280928, A048936, A144563.
%K A020342 nonn,base
%O A020342 1,1
%A A020342 _David W. Wilson_
%E A020342 Edited by _N. J. A. Sloane_, Jan 03 2009