A282150 -id:A282150 - cp's OEIS frontend

A282151 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 10.

11, 22, 33, 44, 55, 66, 77, 88, 99, 102, 110, 113, 124, 135, 146, 157, 168, 179, 201, 204, 215, 220, 226, 237, 248, 259, 306, 311, 317, 328, 330, 339, 402, 408, 419, 421, 440, 512, 531, 550, 603, 622, 641, 660, 713, 732, 751, 770, 804, 823, 842, 861, 880, 914, 933

Offset: 1

Paolo P. Lava, Feb 15 2017

All the palindromic numbers in base 10 with an even number of digits belong to the sequence.

Here the fulcrum is between two digits while in the sequence from A282107 to A282115 is one of the digits.

			If we split 2039 in 203 and 9 we have 3*1 + 0*2 + 2*3 = 9 for the left side and 9*1 = 9 for the right one.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A282107 - A282115, A282143 - A282150.

P:=proc(n,h) local a,j,k: a:=convert(n, base, h):
for k from 1 to nops(a)-1 do
if add(a[j]*(k-j+1),j=1..k)=add(a[j]*(j-k),j=k+1..nops(a))
then RETURN(n); break: fi: od: end: seq(P(i,10),i=1..10^3);

A282149 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 8.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 66, 72, 75, 84, 93, 102, 111, 129, 132, 141, 144, 150, 159, 198, 201, 207, 216, 258, 273, 288, 330, 345, 360, 387, 402, 417, 432, 459, 474, 489, 504, 513, 515, 524, 528, 533, 542, 551, 576, 581, 585, 590, 599, 600, 642, 647, 657, 672

Offset: 1

Paolo P. Lava, Feb 15 2017

All the palindromic numbers in base 8 with an even number of digits belong to the sequence.
Here the fulcrum is between two digits while in the sequence from A282107 to A282115 is one of the digits.
The first number with this property in all the bases from 2 to 8 is
10296444436. - Giovanni Resta, Feb 16 2017

			672 in base 8 is 1240. If we split the number in 12 and 40 we have 2*1 + 1*2 = 4 for the left side and 4*1 + 0*2 = 4 for the right one.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A282107 - A282115, A282143 - A282148, A282150, A282151.

P:=proc(n,h) local a,j,k: a:=convert(n, base, h):
for k from 1 to nops(a)-1 do
if add(a[j]*(k-j+1),j=1..k)=add(a[j]*(j-k),j=k+1..nops(a))
then RETURN(n); break: fi: od: end: seq(P(i,8),i=1..10^3);

cp's OEIS Frontend

A282151 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 10.

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A282149 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 8.

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cp's OEIS Frontend

A282151 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)*d_i} = Sum_{i=1..j-1}{(j-i)*d_i}. Case x = 10.

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Maple

A282149 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)*d_i} = Sum_{i=1..j-1}{(j-i)*d_i}. Case x = 8.

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Maple

A282151 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 10.

A282149 Numbers n with k digits in base x (MSD(n)=d_k, LSD(n)=d_1) such that, chosen one of their digits in position d_k < j < d_1, is Sum_{i=j..k}{(i-j+1)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 8.