A144563 - OEIS

%I #32 Mar 03 2024 03:04:29

%S 1260,1395,1530,6880,12060,12550,12600,13950,14350,15030,15300,18270,

%T 21870,31590,37840,68800,102510,102550,104260,105210,105250,105264,

%U 105750,112590,114390,116928,118440,118480,119682,120060,120600,123840,125050,125460,125500,125950

%N Subset of A020342 (vampire numbers, definition 1) listing numbers which have more than one such representation of the desired form.

%C From _M. F. Hasler_, Nov 01 2021: (Start)

%C If x is in A020342, then 10*x is in this sequence, and this makes up most of the terms. Exceptions are the terms not ending in 0, {1395, 105264, 116928, 119682, 192375, 258795, 263736, 268398, 289674, 1008126, 1133484, 1173939, ...}. There are terms of the form 10*x in A020342 with x not in A020342, like {25510, 45760, 67950, 136590, 146520, 168520, 175560, 246150, 250510, 255010, ...}. Is any such term in this sequence A144563, or can it be proved there isn't?

%C All terms have at least 3 distinct prime factors (omega, A001221), and 4 prime factors counted with multiplicity (bigomega, A001222). The squarefree terms are {132430, 174370, 1012990, 1073290, 1094730, 1156990, 1170670, 117393, ...}. (End)

%H M. F. Hasler, <a href="/A144563/b144563.txt">Table of n, a(n) for n = 1..184</a>, Nov 01 2021

%F Subsequence of A020342; contains 10*A020342 as a subsequence. - _M. F. Hasler_, Nov 01 2021

%e 1260 = 21*60 = 6*210. 1395 = 5*9*31 = 15*93. 1530 = 30*51 = 3*510.

%o (PARI) /* Helper function: count number of representations of n as product of numbers >= m whose multiset of digits is D, excluding the trivial representation if m = 0. */

%o VampRepCount(n, m=0, D=vecsort(digits(n)))={ if(#D<3, m && (D[1]>=m && vecprod(D)==n || vecsort(digits(n))==D), n >= m^2, my(S=Set(D), c=m && vecsort(digits(n))==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 && c += VampRepCount(n\f, f, C) ); c, n >= m && vecsort(digits(n))==D)}

%o is_A144563(n)=VampRepCount(n)>1 \\ _M. F. Hasler_, Nov 01 2021

%Y Closely related: A020342, A014575, A080718, A280928, A048936.

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, Jan 03 2009, based on email from _Zak Seidov_

%E Corrected A-number in definition. More terms and examples _R. J. Mathar_, Jan 05 2009

%E Terms beyond a(15) by _M. F. Hasler_, Nov 01 2021