A282151 -id:A282151 - 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);

A282107 Numbers n with k digits in base x (MSD(n)x=d_k, LSD(n)_x=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 = 2.

Original entry on oeis.org

5, 7, 10, 14, 17, 20, 21, 27, 28, 31, 34, 35, 39, 40, 42, 49, 54, 56, 57, 62, 65, 68, 70, 73, 78, 80, 84, 85, 93, 98, 99, 107, 108, 112, 114, 119, 124, 127, 130, 133, 136, 140, 141, 146, 147, 155, 156, 160, 161, 167, 168, 170, 175, 177, 186, 196, 198, 201, 214

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 2 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.

			897 in base 2 is 1110000001. If j = 7 (the first 0 from left) we have 1*1 + 1*2 + 1*3 = 6 for the left side and 0*1 + 0*2 + 0*3 + 0*4 + 0*5 + 1*6 = 6 for the right one.

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

Cf. A282108 - 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,2),i=1..10^3);

A282143 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 = 2.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 19, 24, 25, 30, 33, 36, 38, 45, 48, 50, 51, 60, 63, 66, 69, 72, 75, 76, 81, 87, 90, 96, 100, 102, 105, 117, 120, 126, 129, 131, 132, 138, 143, 144, 150, 152, 153, 162, 165, 174, 179, 180, 189, 192, 193, 195, 200, 204, 205, 210, 219, 231, 234

Offset: 1

Paolo P. Lava, Giovanni Resta, Feb 07 2017

All the palindromic numbers in base 2 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.

			143 in base 2 is 10001111. If we split the number in 10001 and 111 we have 1*1 + 0*2 + 0*3 + 0*4 + 1*5 = 6 for the left side and 1*1 + 1*2 + 1*3 = 6 for the right one.

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

Cf. A282107 - A282115, A282144 - 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,2),i=1..10^3);

A282108 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 = 3.

Original entry on oeis.org

10, 13, 16, 20, 23, 26, 29, 30, 32, 35, 39, 48, 55, 60, 64, 69, 73, 78, 82, 87, 90, 91, 96, 100, 105, 112, 117, 121, 130, 137, 142, 144, 146, 151, 155, 160, 164, 165, 169, 173, 178, 180, 182, 187, 192, 194, 203, 207, 212, 219, 224, 233, 234, 242, 246, 247, 256

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 3 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.

			35 in base 3 is 1022. If j = 2 (second 2 from the right) we have 0*1 + 1*2 = 2 for the left side and 2*1 for the right one.

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

Cf. A282107, A282109 - 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,3),i=1..10^3);

A282109 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 = 4.

Original entry on oeis.org

17, 21, 25, 29, 34, 38, 42, 46, 51, 55, 59, 63, 66, 68, 70, 74, 78, 83, 84, 87, 91, 95, 100, 116, 129, 136, 145, 152, 161, 168, 177, 184, 197, 204, 213, 220, 229, 236, 245, 252, 257, 259, 263, 264, 267, 271, 272, 273, 280, 289, 296, 305, 312, 325, 332, 336, 341

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 4 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.

			83 in base 4 is 1103. If j = 2 (digit 0) 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

A282107, A282108, A282110 - 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,4),i=1..10^3);

A282110 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 = 5.

Original entry on oeis.org

26, 31, 36, 41, 46, 52, 57, 62, 67, 72, 78, 83, 88, 93, 98, 104, 109, 114, 119, 124, 127, 130, 132, 137, 142, 147, 153, 155, 158, 163, 168, 173, 179, 180, 184, 189, 194, 199, 205, 230, 251, 254, 259, 260, 264, 269, 274, 276, 285, 301, 310, 326, 335, 351, 360, 381

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 5 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 5 are: 78, 650, 1550, 4368, 4433, 4805, 6913, 7410, 16709, 31824, 35175, 41216, 104272, 107584, 132285, 144781, 165059, 173305, 174096, 190468, 195473, 201900, 205005, 205261, 214432, 231521, 243984, 275026, 278528, 295275, 304562, 313769, ...

			137 in base 5 is 1022. If j=2 (the second 2 from right) we have 0*1 + 1*2 = 2 for the left side and 2*1 = 2 for the right one.

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

CF. A282107 - A282109, A282111 - 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,5),i=1..10^3);

A282111 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 = 6.

Original entry on oeis.org

37, 43, 49, 55, 61, 67, 74, 80, 86, 92, 98, 104, 111, 117, 123, 129, 135, 141, 148, 154, 160, 166, 172, 178, 185, 191, 197, 203, 209, 215, 218, 222, 224, 230, 236, 242, 248, 255, 258, 261, 267, 273, 279, 285, 292, 294, 298, 304, 310, 316, 322, 329, 330, 335, 341

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 6 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 6 are:
144781, 345440, 743687, 1650704, 4020912, 4270149, 4757093, 6922591, 7102553, 7406643, 7677171, 7823009, 8853188, 12444016, 14457746, 14853520, 14861718, 15794512, 15994195, 17375742, 20450682, 20802565, 22173561, 22186557, 25268754, 261656297, 26648201, 27740672, ...

			304 in base 6 is 1224. If j = 2 (the first 2 from right) we have 2*1 + 1*2 = 4 for the left side and 4*1 = 4 for the right one.

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

Cf. A282107 - A282110, A282112 - 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,6),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);

A282113 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 = 8.

Original entry on oeis.org

65, 73, 81, 89, 97, 105, 113, 121, 130, 138, 146, 154, 162, 170, 178, 186, 195, 203, 211, 219, 227, 235, 243, 251, 260, 268, 276, 284, 292, 300, 308, 316, 325, 333, 341, 349, 357, 365, 373, 381, 390, 398, 406, 414, 422, 430, 438, 446, 455, 463, 471, 479, 487, 495

Offset: 1

Paolo P. Lava, Feb 06 2017

All the palindromic numbers in base 8 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 8 are: 2438269535, 6936679443, 8657968788, 11107027008, 21733512704, ... - Giovanni Resta, Feb 13 2017

			1084 in base 8 is 2074. If j = 2 (digit 7) we have 0*1 + 2*2 = 4 for the left side and 4*1 = 4 for the right one.

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

Cf. A282107 - A282112, A282114, 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,8),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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A282107 Numbers n with k digits in base x (MSD(n)x=d_k, LSD(n)_x=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 = 2.

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A282143 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 = 2.

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A282108 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 = 3.

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A282109 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 = 4.

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A282110 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 = 5.

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A282111 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 = 6.

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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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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.

A282107 Numbers n with k digits in base x (MSD(n)x=d_k, LSD(n)_x=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 = 2.

A282143 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 = 2.

A282108 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 = 3.

A282109 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 = 4.

A282110 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 = 5.

A282111 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 = 6.

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.

A282113 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 = 8.

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.