A256080 -id:A256080 - cp's OEIS frontend

A256075 Non-palindromic balanced numbers (in base 10).

1030, 1140, 1250, 1302, 1360, 1412, 1470, 1522, 1580, 1603, 1632, 1690, 1713, 1742, 1823, 1852, 1904, 1933, 1962, 2031, 2060, 2141, 2170, 2251, 2280, 2303, 2361, 2390, 2413, 2471, 2523, 2581, 2604, 2633, 2691, 2714, 2743, 2824, 2853, 2905, 2934, 2963, 3032, 3061, 3090, 3142, 3171, 3252, 3281, 3304, 3362, 3391, 3414

Offset: 1

Eric Angelini and 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.
All 1-, 2- or 3-digit balanced numbers are palindromic, therefore all terms are larger than 1000.
The least 1-9 pandigital balanced number seems to be 137986542, but there seems to be no 0-9 pandigital balanced number.

			a(1)=1030 is balanced because 1*3/2 + 0*1/2 = 3*1/2 + 0*3/2.
a(2)=1140 is balanced because 1*3/2 + 1*1/2 = 4*1/2 + 0*3/2.

Robert Israel, Table of n, a(n) for n = 1..10000
E. Angelini, Balanced numbers, SeqFan list, Mar 14 2015

Cf. A256076 (primes in this sequence), A256082 - A256089, A256080.

filter:= proc(n) local L,m;
  L:= convert(n,base,10);
  m:= (1+nops(L))/2;
add(L[i]*(i-m),i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1000..10000]); # Robert Israel, May 29 2018

is(n,b=10,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A256082 Non-palindromic balanced numbers in base 2.

Original entry on oeis.org

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)

A256089 Non-palindromic balanced numbers in base 9.

Original entry on oeis.org

756, 846, 936, 974, 1026, 1064, 1116, 1154, 1206, 1218, 1244, 1308, 1334, 1398, 1424, 1486, 1512, 1576, 1602, 1666, 1692, 1704, 1756, 1794, 1846, 1884, 1936, 1948, 1974, 2038, 2064, 2128, 2154, 2216, 2242, 2306, 2332, 2396, 2422, 2434, 2486, 2524, 2576, 2614, 2666, 2678, 2704, 2768, 2794, 2858, 2884, 2946, 2972

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. Since palindromes (A029955) are trivially balanced, they are excluded here.

This is the base-9 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

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

Cf. A256082 - A256088, A256075, A256080, A029955.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 9);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=9,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A256083 Non-palindromic balanced numbers in base 3.

Original entry on oeis.org

87, 96, 105, 137, 146, 155, 169, 178, 187, 264, 276, 312, 348, 380, 416, 452, 464, 508, 520, 556, 592, 741, 768, 795, 816, 831, 843, 858, 870, 885, 895, 906, 922, 933, 949, 960, 987, 991, 1014, 1018, 1041, 1045, 1055, 1077, 1082, 1104, 1109, 1131, 1141, 1145, 1168, 1172, 1195, 1199, 1226, 1237, 1253, 1264, 1280, 1291, 1301

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 base-3 variant of the decimal version A256075 invented by Eric Angelini.
All balanced numbers with less than 4 digits are palindromic, and since there is no digit 3 in base 3, there cannot be a term in this sequence with 4 base-3 digits, where weights are (-3/2, -1/2, 1/2, 3/2).

			a(4) = 137 = 12002[3] is balanced because 1*2 + 2*1 = 0*1 + 2*2.

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

Cf. A256082 - A256089, A256075, A256080.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 3);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=3,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A256088 Non-palindromic balanced numbers in base 8.

Original entry on oeis.org

536, 608, 680, 706, 752, 778, 824, 850, 899, 922, 971, 994, 1049, 1072, 1121, 1144, 1193, 1219, 1265, 1291, 1337, 1363, 1412, 1435, 1484, 1507, 1562, 1585, 1634, 1657, 1706, 1732, 1778, 1804, 1850, 1876, 1925, 1948, 1997

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. Since palindromes (A029803) are trivially balanced, they are excluded here.

This is the base-8 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

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

Cf. A256082 - A256089, A256075, A256080, A029803.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 8);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=8,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A256084 Non-palindromic balanced numbers in base 4.

Original entry on oeis.org

76, 114, 141, 179, 206, 264, 280, 296, 312, 332, 348, 364, 380, 386, 402, 418, 434, 454, 470, 486, 502, 521, 537, 553, 569, 589, 605, 621, 637, 643, 659, 675, 691, 711, 727, 743, 759, 778, 794, 810, 826, 846, 862, 878, 894, 1060, 1096, 1140, 1176, 1221, 1256, 1320, 1329, 1356, 1400, 1410, 1436, 1481, 1490, 1516, 1554, 1580

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 base-4 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

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

Cf. A256082 - A256089, A256075, A256080.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 4);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=4,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A256085 Non-palindromic balanced numbers in base 5.

Original entry on oeis.org

140, 170, 202, 232, 266, 296, 328, 358, 392, 422, 454, 484, 518, 548, 635, 660, 685, 710, 735, 765, 790, 815, 840, 865, 877, 895, 902, 920, 927, 945, 952, 970, 977, 995, 1007, 1032, 1057, 1082, 1107, 1128, 1137, 1153, 1162, 1178, 1187, 1203, 1212, 1228, 1237, 1261, 1270, 1286, 1295, 1311, 1320, 1336, 1345, 1361, 1370

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. Since palindromes (A029952) are trivially balanced, they are excluded here.

This is the base-5 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

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

Cf. A256082 - A256089, A256075, A256080, A029952.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 5);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=5,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A256086 Non-palindromic balanced numbers in base 6.

Original entry on oeis.org

234, 276, 318, 326, 368, 410, 451, 493, 535, 543, 585, 627, 668, 710, 752, 760, 802, 844, 885, 927, 969, 977, 1019, 1061, 1102, 1144, 1186, 1308, 1344, 1380, 1416, 1452, 1488, 1530, 1566, 1602, 1638, 1674, 1710, 1730, 1752, 1766, 1788, 1802, 1824, 1838, 1860, 1874, 1896, 1910, 1932, 1952, 1974, 1988

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. Since palindromes (A029953) are trivially balanced, they are excluded here.

This is the base-6 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

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

Cf. A256082 - A256089, A256075, A256076, A256080, A029953.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 6);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=6,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&d!=Vecrev(d)

A256087 Non-palindromic balanced numbers in base 7.

Original entry on oeis.org

364, 420, 476, 492, 532, 548, 604, 640, 660, 708, 728, 764, 820, 836, 876, 892, 948, 984, 1004, 1052, 1072, 1108, 1164, 1180, 1220, 1236, 1292, 1328, 1348, 1396, 1416, 1452, 1508, 1524, 1564, 1580, 1636, 1672, 1692, 1740, 1760, 1796, 1852, 1868, 1908, 1924, 1980

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. Since palindromes (A029954) are trivially balanced, they are excluded here.

This is the base-7 variant of the decimal version A256075 invented by Eric Angelini. See there, and the base-2 version A256082, for further information and examples.

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

Cf. A256082 - A256089, A256075, A256080, A029954.

filter:= proc(n) local L, m,i;
  L:= convert(n, base, 7);
  m:= (1+nops(L))/2;
add(L[i]*(i-m), i=1..nops(L))=0  and L <> ListTools:-Reverse(L)
end proc:
select(filter, [$1..10000]); # Robert Israel, Nov 04 2024

is(n,b=7,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)

A256090 Non-palindromic balanced primes in base 16.

Original entry on oeis.org

6451, 7717, 8513, 8963, 9601, 10501, 10867, 11317, 11411, 12227, 13829, 14561, 15461, 15733, 16183, 16529, 16979, 18517, 19333, 19427, 19699, 20149, 20233, 20327, 22483, 22567, 23027, 23561, 23833, 25717, 26083, 26261, 26711, 27077, 27527, 27799, 27893, 28867, 29411, 29683, 30133, 30677, 30949, 31033, 31849

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. Palindromic primes (A029732 in base 16) are trivially balanced, therefore they are excluded here.

These are the primes in A256080. This is the hexadecimal variant of the decimal version A256076 suggested by Eric Angelini.

Cf. A256080, A256076, A256075, A256082 - A256089, A029732.

is(n,b=16,d=digits(n,b),o=(#d+1)/2)=!(vector(#d,i,i-o)*d~)&&d!=Vecrev(d)&&isprime(n)

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A256075 Non-palindromic balanced numbers (in base 10).

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A256082 Non-palindromic balanced numbers in base 2.

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A256089 Non-palindromic balanced numbers in base 9.

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A256083 Non-palindromic balanced numbers in base 3.

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A256088 Non-palindromic balanced numbers in base 8.

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A256084 Non-palindromic balanced numbers in base 4.

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A256085 Non-palindromic balanced numbers in base 5.

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A256086 Non-palindromic balanced numbers in base 6.

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