A282113 -id:A282113 - cp's OEIS frontend

A282115 Numbers m with k digits in base b (MSD(m)=d_k, LSD(m)=d_1) such that, for one of their digits in position d_k < j < d_1, Sum_{i=j+1..k} (i-j)d_i = Sum_{i=1..j-1} (j-i)d_i. Case b = 10.

101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626

Offset: 1

Paolo P. Lava, Feb 06 2017

All the numbers that are palindromic in base 10 and have an odd number of digits belong to this sequence.

Here the fulcrum is one of the digits while in the sequences from A282143 to A282151 it is between two digits.

			10467: if j = 2 (digit 6) we have 4*1 + 0*2 + 1*3 = 7 for the left side and 7*1 = 7 for the right side.

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

Cf. A282107, A282108, A282109, A282110, A282111, A282112, A282113, A282114.

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),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);

A282112 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+1..k}{(i-j)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 7.

Original entry on oeis.org

50, 57, 64, 71, 78, 85, 92, 100, 107, 114, 121, 128, 135, 142, 150, 157, 164, 171, 178, 185, 192, 200, 207, 214, 221, 228, 235, 242, 250, 257, 264, 271, 278, 285, 292, 300, 307, 314, 321, 328, 335, 342, 345, 350, 352, 359, 366, 373, 380, 387, 395, 399, 402, 409

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 7 with an odd number of digits belong to the sequence.
Here the fulcrum is one of the digits while in the sequence from A282143 to A282151 is between two digits.
Numbers with this property in all the bases from 2 to 7 are: 53060873, 55161152, 151009636, 343518281, 505587488, 513015908, ...- Giovanni Resta, Feb 13 2017

			409 in base 7 is 1123. If j = 2 (digit 2) we have 1*1 + 1*2 = 3 for the left side and 3*1 = 3 for the right one.

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

Cf. A282107 - A282111, A282113 - A282115.

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),j=1..k)=add(a[j]*(j-k),j=k+1..nops(a)) then
RETURN(n); break: fi: od: end: seq(P(i,7),i=1..10^3);

A282114 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+1..k}{(i-j)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 9.

Original entry on oeis.org

82, 91, 100, 109, 118, 127, 136, 145, 154, 164, 173, 182, 191, 200, 209, 218, 227, 236, 246, 255, 264, 273, 282, 291, 300, 309, 318, 328, 337, 346, 355, 364, 373, 382, 391, 400, 410, 419, 428, 437, 446, 455, 464, 473, 482, 492, 501, 510, 519, 528, 537, 546, 555

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 9 with an odd number of digits belong to the sequence.
Here the fulcrum is one of the digits while in the sequence from A282143 to A282151 is between two digits.
Numbers with this property in all the bases from 2 to 9 are:
898958160865, 1518029154732,... - Giovanni Resta, Feb 13 2017

			3485 in base 9 is 4702. If j = 3 (digit 7) we have 4*1 = 4 for tyhe left side and 0*1 + 2*2 = 4 for the right one.

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

Cf. A282107 - A282113, A282115.

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),j=1..k)=add(a[j]*(j-k),j=k+1..nops(a)) then
RETURN(n); break: fi: od: end: seq(P(i,9),i=1..10^3);

cp's OEIS Frontend

A282115 Numbers m with k digits in base b (MSD(m)=d_k, LSD(m)=d_1) such that, for one of their digits in position d_k < j < d_1, Sum_{i=j+1..k} (i-j)d_i = Sum_{i=1..j-1} (j-i)d_i. Case b = 10.

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A282112 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+1..k}{(i-j)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 7.

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A282114 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+1..k}{(i-j)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 9.

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cp's OEIS Frontend

A282115 Numbers m with k digits in base b (MSD(m)=d_k, LSD(m)=d_1) such that, for one of their digits in position d_k < j < d_1, Sum_{i=j+1..k} (i-j)*d_i = Sum_{i=1..j-1} (j-i)*d_i. Case b = 10.

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A282112 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+1..k}{(i-j)*d_i} = Sum_{i=1..j-1}{(j-i)*d_i}. Case x = 7.

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A282114 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+1..k}{(i-j)*d_i} = Sum_{i=1..j-1}{(j-i)*d_i}. Case x = 9.

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A282115 Numbers m with k digits in base b (MSD(m)=d_k, LSD(m)=d_1) such that, for one of their digits in position d_k < j < d_1, Sum_{i=j+1..k} (i-j)d_i = Sum_{i=1..j-1} (j-i)d_i. Case b = 10.

A282112 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+1..k}{(i-j)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 7.

A282114 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+1..k}{(i-j)d_i} = Sum_{i=1..j-1}{(j-i)d_i}. Case x = 9.