cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A282144 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 = 3.

Original entry on oeis.org

4, 8, 11, 12, 19, 24, 28, 33, 36, 40, 47, 52, 56, 57, 61, 68, 72, 80, 84, 85, 92, 97, 99, 104, 108, 109, 113, 120, 125, 141, 145, 156, 168, 170, 171, 172, 183, 193, 204, 208, 216, 218, 229, 240, 244, 245, 250, 252, 255, 257, 269, 276, 278, 280, 291, 297, 301, 312
Offset: 1

Views

Author

Paolo P. Lava, Giovanni Resta, Feb 07 2017

Keywords

Comments

All the palindromic numbers in base 3 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.

Examples

			229 in base 3 is 22111. If we split the number in 22 and 111 we have 2*1 + 2*2 = 6 for the left side and 1*1 + 1*2 + 1*3 = 6 for the right one.

Links

Crossrefs

Cf. A282107 - A282115, A282143, A282145 - A282151.

Programs

  • Maple
    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,3),i=1..10^3);

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.

Original entry on oeis.org

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

Views

Author

Paolo P. Lava, Giovanni Resta, Feb 07 2017

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A282107 - A282115, A282143, A282144, A282146 - A282151.

Programs

  • Maple
    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);

A282146 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 = 5.

Original entry on oeis.org

6, 12, 18, 24, 27, 30, 33, 39, 51, 54, 60, 81, 90, 102, 111, 120, 126, 128, 134, 135, 150, 156, 165, 177, 186, 195, 207, 216, 228, 237, 246, 252, 255, 261, 270, 282, 291, 300, 303, 312, 321, 333, 342, 354, 363, 372, 376, 378, 387, 396, 405, 408, 417, 429, 438, 447
Offset: 1

Views

Author

Paolo P. Lava, Giovanni Resta, Feb 07 2017

Keywords

Comments

All the palindromic numbers in base 5 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 5 are: 3120, 9615, 10366, 16610, 17812, 22129, 33329, 100726, 163800, 202039, 208172, 212636, 258221, 270337, 298575, 420240, 462608, 475782, 492420, 523679, 549537, 550200, 587842, 594511, 610273, 655350, 671844, 675872, 681280, 730161, 738480, 840798, 842614, 848655, 855973, 925515, 987751, ...

Examples

			447 in base 5 is 3242. If we split the number in 32 and 42 we have 2*1 + 3*2 = 8 for the left side and 4*1 + 2*2 = 8 for the right one.

Links

Crossrefs

Cf. A282107 - A282115, A282143 - A282145, A282147 - A282151.

Programs

  • Maple
    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,5),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

Views

Author

Paolo P. Lava, Giovanni Resta, Feb 07 2017

Keywords

Comments

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, ...

Examples

			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.

Links

Crossrefs

Cf. A282107 - A282115, A282143 - A282146, A282148 - A282151.

Programs

  • Maple
    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);

A282148 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 = 7.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 51, 56, 59, 67, 75, 83, 99, 102, 110, 112, 118, 153, 155, 168, 198, 211, 224, 254, 267, 280, 297, 310, 323, 336, 344, 346, 354, 357, 362, 370, 392, 397, 400, 405, 413, 443, 456, 469, 499, 512, 525, 542, 555, 568, 581, 598, 611, 624, 641, 654
Offset: 1

Views

Author

Paolo P. Lava, Feb 15 2017

Keywords

Comments

All the palindromic numbers in base 7 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 7 is
86964945. - Giovanni Resta, Feb 15 2017

Examples

			641 in base 7 is 1604. If we split the number in 16 and 04 we have 6*1 + 1*2 = 8 for the left side and 0*1 + 4*2 = 8 for the right one.

Links

Crossrefs

Cf. A282107 - A282115, A282143 - A282147.

Programs

  • Maple
    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,7),i=1..10^3);

A282150 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 = 9.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 83, 90, 93, 103, 113, 123, 133, 143, 163, 166, 176, 180, 186, 196, 206, 249, 253, 259, 269, 270, 326, 332, 343, 360, 416, 433, 450, 489, 506, 523, 540, 579, 596, 613, 630, 652, 669, 686, 703, 720, 730, 732, 742, 747, 752, 762, 772
Offset: 1

Views

Author

Paolo P. Lava, Feb 15 2017

Keywords

Comments

All the palindromic numbers in base 9 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 9 is 1030854453981. - Giovanni Resta, Feb 16 2017

Examples

			762 in base 9 is 1036. If we split the number in 103 and 6 we have 3*1 + 0*2 + 1*3 = 6 for the left side and 6*1 = 6 for the right one.

Links

Crossrefs

Cf. A282107-A282115, A282143-A282149, A282151.

Programs

  • Maple
    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,9),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

Views

Author

Paolo P. Lava, Feb 15 2017

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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);

A373117 Stable numbers on vertical blade (see the Example section for an explanation).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 102, 110, 111, 113, 121, 124, 131, 135, 141, 146, 151, 157, 161, 168, 171, 179, 181, 191, 200, 201, 202, 204, 212, 215, 220, 222, 226, 232, 237, 242, 248, 252, 259, 262, 272, 282, 292, 300, 303, 306, 311
Offset: 1

Views

Author

Eric Angelini and Giorgos Kalogeropoulos, May 25 2024

Keywords

Examples

			We place the k digits of a number in succession in k adjacent square boxes forming a rectangle of base k and height 1. We can only place a vertical blade under this rectangle in two ways: exactly between two boxes [positions (a) and (d) below] or exactly under the middle of a box [position (b) and (c) below].
If the blade is placed in position (a) between the two 1s of 11, the number 11 is stable.
If the blade is placed in (b) exactly under the 0 of 101, the number 101 is stable.
If the blade is placed in (c) exactly under the 1 of 10, the number is stable.
If the blade is placed in (d) between the 0 and the 2 of 102, the number 102 is stable.
For the last two examples, the balance of the number considered can be explained as follows: the distance which separates a digit from the blade comes into play - the more this distance increases, the heavier the digit in question is. So, the digit 1 of 102 weighs 2 in reality (weight*distance = 1*2 = 2). This quantity balances the influence of the 2 of 102 (for which weight*distance = 2*1 = 2 too).
.
.+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
.|   | 1 | 1 |   | 1 | 0 | 1 |   | 1 | 0 |   | 1 | 0 | 2 |   |
.+---+---a---+---+---+-b-+---+---+-c-+---+---+---+---d---+---+
.|   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
.+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
.

Links

Crossrefs

Cf. A282115, A282151.
Previous Showing 11-18 of 18 results.

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