A144563 Subset of A020342 (vampire numbers, definition 1) listing numbers which have more than one such representation of the desired form.

1260, 1395, 1530, 6880, 12060, 12550, 12600, 13950, 14350, 15030, 15300, 18270, 21870, 31590, 37840, 68800, 102510, 102550, 104260, 105210, 105250, 105264, 105750, 112590, 114390, 116928, 118440, 118480, 119682, 120060, 120600, 123840, 125050, 125460, 125500, 125950

Offset

1,1

Comments

From M. F. Hasler, Nov 01 2021: (Start)

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?

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)

Links

M. F. Hasler, Table of n, a(n) for n = 1..184, Nov 01 2021

Formula

Subsequence of A020342; contains 10*A020342 as a subsequence. - M. F. Hasler, Nov 01 2021

Example

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

Prog

(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. */
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)}
is_A144563(n)=VampRepCount(n)>1 \\ M. F. Hasler, Nov 01 2021

Crossrefs

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

Sequence in context: A099592 A240922 A014575 * A175746 A291714 A350375

Adjacent sequences: A144560 A144561 A144562 * A144564 A144565 A144566

Keyword

nonn,base

Author

N. J. A. Sloane, Jan 03 2009, based on email from Zak Seidov

Extensions

Corrected A-number in definition. More terms and examples R. J. Mathar, Jan 05 2009

Terms beyond a(15) by M. F. Hasler, Nov 01 2021

Status

approved