A282146 -id:A282146 - cp's OEIS frontend

A282145 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 = 4.

5, 10, 15, 18, 20, 23, 33, 40, 53, 60, 65, 67, 72, 80, 85, 92, 98, 105, 118, 125, 130, 132, 137, 150, 157, 160, 163, 170, 183, 190, 193, 195, 202, 212, 215, 222, 235, 240, 255, 260, 261, 268, 274, 281, 288, 294, 301, 307, 314, 320, 321, 326, 333, 339, 340, 346

Offset: 1

Paolo P. Lava, Giovanni Resta, Feb 07 2017

All the palindromic numbers in base 4 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.

			222 in base 4 is 3132. If we split the number in 31 and 32 we have 1*1 + 3*2 = 7 for the left side and 3*1 + 2*2 = 7 for the right one.

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

Cf. A282107 - A282115, A282143, A282144, A282146 - 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,4),i=1..10^3);

A282147 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 = 6.

Original entry on oeis.org

7, 14, 21, 28, 35, 38, 42, 45, 52, 59, 73, 76, 83, 84, 115, 126, 146, 157, 168, 188, 199, 210, 217, 219, 226, 228, 233, 252, 257, 259, 270, 290, 301, 312, 332, 343, 354, 363, 374, 385, 405, 416, 427, 434, 438, 445, 456, 476, 487, 498, 504, 507, 518, 529, 549, 560

Offset: 1

Paolo P. Lava, Giovanni Resta, Feb 07 2017

All the palindromic numbers in base 6 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.
Numbers with this property in all the bases from 2 to 6 are:
420240, 610273, 848655, 855973, 987751, 1830038, 2347657, 3480366, 3519545, 4832865, 5141958, 6050107, 9010530, 9770426, 11520023, 13951022, 14036167, 14694080, 15106072, 16487203, 24125707, 25209012, ...

			580 in base 6 is 2404. If we split the number in 24 and 04 we have 4*1 + 2*2 = 8 for the left side and 0*1 + 4*2 = 8 for the right one.

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

Cf. A282107 - A282115, A282143 - A282146, A282148 - 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,6),i=1..10^3);

cp's OEIS Frontend

A282145 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 = 4.

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A282147 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 = 6.

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cp's OEIS Frontend

A282145 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 = 4.

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Maple

A282147 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 = 6.

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Maple

A282145 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 = 4.

A282147 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 = 6.