A256083 -id:A256083 - cp's OEIS frontend

A256082 Non-palindromic balanced numbers in base 2.

70, 78, 150, 266, 282, 294, 310, 334, 350, 355, 371, 397, 413, 540, 554, 582, 630, 686, 723, 798, 813, 1036, 1042, 1068, 1074, 1098, 1116, 1130, 1148, 1158, 1178, 1190, 1210, 1221, 1238, 1253, 1270, 1302, 1305, 1334, 1337, 1347, 1358, 1379, 1390, 1427, 1438, 1459, 1470, 1483, 1515, 1550, 1557, 1582, 1589, 1613, 1630

Offset: 1

M. F. Hasler, Mar 14 2015

Here a number is called balanced if the sum of digits weighted by their arithmetic distance from the "center" is zero.
This is the binary variant of the base-10 version A256075 invented by Eric Angelini. See A256081 for the primes in this sequence. See A256083 - A256089 and A256080 for variants in other bases.
If n is in the sequence with 2^d < n < 2^(d+1), then 2^(d+2)+2*n+1 is in the sequence, as are n*(2^k+1) for k > d. - Robert Israel, May 29 2018

			a(1) = 70 = 1000110[2] is balanced because 1*3 = 1*1 + 1*2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A256081 (primes), A256083, A256084, A256085, A256086, A256087, A256088, A256089, A256075.

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

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

A330491 Non-palindromic balanced primes in base 3.

Original entry on oeis.org

137, 991, 1109, 1237, 1291, 1301, 1471, 1663, 1721, 1861, 1871, 7057, 7219, 7507, 7537, 7699, 8291, 8597, 8707, 9091, 9587, 9697, 9857, 10159, 10163, 10211, 10273, 10321, 10627, 10631, 10739, 11027, 11437, 11551, 11777, 11887, 12239, 12401, 12659, 12671, 12821

Offset: 1

Thorben Böger, Dec 16 2019

A number is called "balanced" here if the sum of digits weighted by their arithmetic distance from the "center" of the number is zero. Palindromic primes (A029971) are "trivially" balanced, so they are excluded here.

These are the primes in A256083, respectively the intersection of A000040 and A256083.

			a(7) = 1471 as 1471 is prime and 2000111 in base 3, which is balanced: 3*2 = 1*1 + 2*1 + 3*1.

Thorben Böger, Table of n, a(n) for n = 1..19999

Cf. A029971, A256083, A256076, A000040.

ok(n)={my(v=digits(n,3)); isprime(n) && !sum(i=1, #v, v[i]*((#v+1)/2-i)) && Vecrev(v)<>v} \\ Andrew Howroyd, Dec 23 2019

from primes_file import primes#list containing first 3 million primesfrom baseconvert import base as bdef isbalanced(converted):    return sum([(place - (len(converted)/2 - 0.5))*digit for place, digit in enumerate(converted)]) == 0balanced_primes_list = [prime for prime in primes if(b(prime, 10, 3) != b(prime, 10, 3)[::-1] and isbalanced(b(prime, 10, 3)))]

cp's OEIS Frontend

A256082 Non-palindromic balanced numbers in base 2.

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