A256075 Non-palindromic balanced numbers (in base 10).

1030, 1140, 1250, 1302, 1360, 1412, 1470, 1522, 1580, 1603, 1632, 1690, 1713, 1742, 1823, 1852, 1904, 1933, 1962, 2031, 2060, 2141, 2170, 2251, 2280, 2303, 2361, 2390, 2413, 2471, 2523, 2581, 2604, 2633, 2691, 2714, 2743, 2824, 2853, 2905, 2934, 2963, 3032, 3061, 3090, 3142, 3171, 3252, 3281, 3304, 3362, 3391, 3414

Offset: 1

Author

Eric Angelini and M. F. Hasler, Mar 14 2015

Keywords

• nonn
• base

Comments

Here a number is called balanced if the sum of digits weighted by their arithmetic distance from the "center" is zero.

All 1-, 2- or 3-digit balanced numbers are palindromic, therefore all terms are larger than 1000.

The least 1-9 pandigital balanced number seems to be 137986542, but there seems to be no 0-9 pandigital balanced number.

Examples

			a(1)=1030 is balanced because 1*3/2 + 0*1/2 = 3*1/2 + 0*3/2.
a(2)=1140 is balanced because 1*3/2 + 1*1/2 = 4*1/2 + 0*3/2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• E. Angelini, Balanced numbers, SeqFan list, Mar 14 2015

Crossrefs

Cf. A256076 (primes in this sequence), A256082 - A256089, A256080.

Programs

Maple

filter:= proc(n) local L,m;
  L:= convert(n,base,10);
  m:= (1+nops(L))/2;
add(L[i]*(i-m),i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1000..10000]); # Robert Israel, May 29 2018

PARI

is(n,b=10,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)