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A014575 Vampire numbers (definition 2): numbers n with an even number of digits which have a factorization n = i*j where i and j have the same number of digits and the multiset of the digits of n coincides with the multiset of the digits of i and j.

%I A014575 #84 Mar 14 2025 19:01:11
%S A014575 1260,1395,1435,1530,1827,2187,6880,102510,104260,105210,105264,
%T A014575 105750,108135,110758,115672,116725,117067,118440,120600,123354,
%U A014575 124483,125248,125433,125460,125500,126027,126846,129640
%N A014575 Vampire numbers (definition 2): numbers n with an even number of digits which have a factorization n = i*j where i and j have the same number of digits and the multiset of the digits of n coincides with the multiset of the digits of i and j.
%C A014575 The numbers i and j may not both have trailing zeros. Numbers may have more than one such factorization. However, each n is listed only once. [Comment modified by _Rick L. Shepherd_, Nov 02 2009]
%D A014575 C. A. Pickover, "Vampire Numbers." Ch. 30 in Keys to Infinity. New York: Wiley, pp. 227-231, 1995.
%H A014575 Giovanni Resta, <a href="/A014575/b014575.txt">Table of n, a(n) for n = 1..10000</a> (terms a(1)-a(87) by R. J. Mathar and a(88)-a(1006) by Manfred Scheucher)
%H A014575 Ely Golden, <a href="/A014575/a014575.py.txt">Sympy program for generating vampire numbers (definition 2)</a>
%H A014575 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 A014575 Manfred Scheucher, <a href="/A014575/a014575.sage.txt">Sage Script</a>
%H A014575 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VampireNumber.html">Vampire Number</a>
%e A014575 1260 = 21*60, 1395 = 15*93, 1435 = 35*41, 1530 = 30*51, etc.
%p A014575 n := 1 :
%p A014575 for dgs from 4 to 10 by 2 do
%p A014575     for a from 10^(dgs-1) to 10^dgs-1 do
%p A014575         amset := sort(convert(a,base,10)) ;
%p A014575         isv := false ;
%p A014575         for d in numtheory[divisors](a) do
%p A014575             m := a/d ;
%p A014575             if ( m >= d ) then
%p A014575                 dset := convert(d,base,10) ;
%p A014575                 mset := convert(m,base,10) ;
%p A014575                 fset := sort([op(dset),op(mset)]) ;
%p A014575                 if fset = amset and nops(dset) = nops(mset) then
%p A014575                     if (m mod 10 <> 0 ) or (d mod 10 <> 0 ) then
%p A014575                     printf("%d %d\n",n,a) ;
%p A014575                     isv := true ;
%p A014575                     n := n+1 ;
%p A014575                     end if;
%p A014575                 end if;
%p A014575             end if;
%p A014575             if isv then
%p A014575                 break;
%p A014575             end if;
%p A014575         end do:
%p A014575     end do:
%p A014575 end do: # _R. J. Mathar_, Jan 10 2013
%t A014575 fQ[n_] := If[OddQ@ IntegerLength@ n, False, MemberQ[Map[Sort@ Flatten@ IntegerDigits@ # &, Select[Map[{#, n/#} &, TakeWhile[Divisors@ n, # <= Sqrt@ n &]], SameQ @@ Map[IntegerLength, #] &]], Sort@ IntegerDigits@ n]]; Select[Range[10^6], fQ] (* _Michael De Vlieger_, Jan 27 2017 *)
%o A014575 (PARI) is(n)=my(v=digits(n));if(#v%2,return(0));fordiv(n,d,if(#Str(d)==#v/2 && #Str(n/d)==#v/2 && vecsort(v)==vecsort(digits(eval(Str(d,n/d)))) && (d%10 || (n/d)%10), return(1)));0 \\ _Charles R Greathouse IV_, Apr 19 2013
%o A014575 (PARI) is_A014575(n)={my(v=vecsort(Vecsmall(Str(n)))); #v%2 && return; my( M=10^(#v\2), L=M\10); fordiv(n,d, d<L && next; d<M || return; v==vecsort(Vecsmall(Str(d,n/d))) && return(d))} \\ Twice as fast. Returns smallest factor (A048933) if vampire number, or false (empty, 0) else. - _M. F. Hasler_, Mar 11 2021
%Y A014575 The following sequences are all closely related: A020342, A014575, A080718, A280928, A048936, A144563.
%Y A014575 Cf. A048933, A048934, A048935, A048936, A048937, A048938, A048939.
%K A014575 nonn,base
%O A014575 1,1
%A A014575 _Eric W. Weisstein_
%E A014575 Edited by _N. J. A. Sloane_, Jan 03 2009