A023108 -id:A023108 - cp's OEIS frontend

A090062 There is (presumably) one and only one palindrome in the Reverse and Add! trajectory of n.

89, 98, 167, 187, 266, 286, 365, 385, 479, 563, 578, 583, 662, 677, 682, 749, 761, 776, 779, 781, 829, 860, 869, 875, 880, 899, 928, 947, 968, 974, 977, 998, 1077, 1093, 1098, 1167, 1183, 1188, 1257, 1273, 1278, 1297, 1347, 1363, 1368, 1387, 1396, 1397, 1437

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 2000 the only palindrome is reached from the start in at most 24 steps; thereafter no further palindrome is reached in 2000 steps.

			The trajectory of 479 begins 479, 1453, 4994, 9988, 18887, ...; at 9988 it joins the (presumably) palindrome-free trajectory of A063048(3) = 1997, hence 4994 is the only palindrome in the trajectory of 479 and 479 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A090063 Numbers n such that there are (presumably) two palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

49, 58, 67, 76, 85, 94, 108, 118, 127, 133, 143, 148, 153, 173, 177, 178, 198, 207, 217, 226, 239, 247, 276, 277, 279, 297, 306, 316, 325, 331, 338, 339, 341, 346, 349, 351, 371, 375, 376, 378, 379, 396, 405, 415, 419, 430, 437, 438, 440, 445, 448, 450, 464

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 118 begins 118, 929, 1858, 10439, 103840, 152141, 293392, 586784, 1074469, ...; at 1074469 it joins the (presumably) palindrome-free trajectory of A063048(72) = 90379, hence 929 and 293392 are the two palindromes in the trajectory of 118 and 118 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A090064 Numbers n such that there are (presumably) three palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 69, 72, 78, 81, 87, 90, 96, 99, 113, 125, 126, 128, 137, 146, 149, 156, 157, 162, 163, 165, 168, 169, 172, 175, 180, 183, 188, 189, 193, 194, 195, 197, 220, 224, 225, 227, 232, 236, 242, 245, 248, 252, 255, 256, 259, 261, 264, 267, 268

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 113 begins 113, 424, 848, 1696, 8657, 16225, 68486, 136972, 416603, ...; at 416603 it joins the (presumably) palindrome-free trajectory of A063048(16) = 10735, hence 424, 848 and 68486 are the three palindromes in the trajectory of 113 and 113 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A090065 Numbers n such that there are (presumably) four palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

9, 19, 28, 29, 37, 38, 39, 46, 47, 48, 56, 57, 64, 65, 73, 74, 75, 82, 83, 84, 91, 92, 93, 110, 112, 121, 124, 132, 134, 135, 136, 138, 144, 147, 155, 164, 166, 174, 182, 186, 190, 192, 211, 212, 219, 223, 229, 230, 231, 233, 234, 235, 237, 240, 243, 246, 249

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 134 begins 134, 565, 1130, 1441, 2882, 5764, 10439, 103840, 152141, 293392, 586784, 1074469, ...; at 1074469 it joins the (presumably) palindrome-free trajectory of A063048(72) = 90379, hence 565, 1441, 2882 and 293392 are the four palindromes in the trajectory of 134 and 134 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A090066 Numbers n such that there are (presumably) five palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

14, 15, 23, 24, 32, 41, 42, 50, 51, 55, 60, 66, 79, 97, 105, 106, 107, 119, 120, 123, 129, 130, 131, 140, 141, 152, 159, 161, 171, 176, 179, 181, 184, 185, 199, 204, 205, 206, 218, 228, 251, 258, 269, 275, 278, 283, 284, 290, 298, 304, 305, 317, 319, 321, 327

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 106 begins 106, 707, 1414, 5555, 11110, 12221, 24442, 48884, 97768, ...; at 97768 it joins the (presumably) palindrome-free trajectory of A063048(3) = 1997, hence 707, 5555, 12221, 24442 and 48884 are the five palindromes in the trajectory of 106 and 106 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A090067 Numbers n such that there are (presumably) six palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

7, 12, 17, 21, 26, 30, 33, 35, 53, 59, 62, 68, 71, 80, 86, 88, 95, 102, 103, 109, 114, 117, 142, 150, 154, 170, 191, 201, 208, 209, 210, 213, 216, 222, 241, 253, 300, 301, 303, 307, 308, 312, 315, 329, 340, 352, 359, 383, 389, 400, 404, 406, 407, 411, 428, 451

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 154 begins 154, 605, 1111, 2222, 4444, 8888, 17776, 85547, 160105, 661166, 1322332, 3654563, 7309126, ...; at 7309126 it joins the (presumably) palindrome-free trajectory of A063048(7) = 10577, hence 525, 1551, 5115, 13431, 26862 and 12455421 are the six palindromes in the trajectory of 154 and 154 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A090068 Numbers n such that there are (presumably) seven palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

6, 13, 16, 25, 31, 34, 40, 43, 44, 52, 61, 70, 77, 104, 111, 115, 145, 158, 200, 202, 203, 214, 244, 250, 257, 302, 356, 399, 401, 412, 414, 442, 455, 498, 500, 505, 511, 519, 529, 541, 554, 597, 610, 618, 626, 628, 640, 653, 656, 686, 752, 795, 797, 816, 826

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 2000 each palindrome is reached from the preceding one or from the start in at most 24 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 25 begins 25, 77, 154, 605, 1111, 2222, 4444, 8888, 17776, 85547, 160105, 661166, 1322332, 3654563,7309126, ...; at 7309126 it joins the (presumably) palindrome-free trajectory of A063048(7) = 10577, hence 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the seven palindromes in the trajectory of 25 and 25 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594.

A160078 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in binary expansion reversed). Binary analog of Lychrel numbers.

Original entry on oeis.org

22, 26, 28, 35, 37, 41, 45, 46, 47, 49, 60, 61, 67, 75, 77, 78, 84, 86, 89, 90, 93, 94, 95, 97, 105, 106, 108, 110, 116, 120, 122, 124, 125, 131, 135, 139, 141, 147, 149, 152

Offset: 1

Dremov Dmitry (dremovd(AT)gmail.com), May 01 2009

Number of iterations equals 1000, but all non-seeded numbers (under) fall out in 32 iterations

			22 = 10110
10110 + 01101 = 100011
100011 + 110001 = 1010100...
Not forming a palindrome after 1000 iterations.

Diofant.ru, Problem: binary Lychrel numbers under 1024. (Russian language!) [From Dremov Dmitry (dremovd(AT)gmail.com), May 03 2009]

Binary version of A023108.

from sympy.ntheory.digits import digits
def make_int(l, b):
    return int(''.join(str(d) for d in l), b)
maxn = 102
it = []
for i in range( 1, maxn ) :
    d = digits( i, 2 )[1:]
    isLychrel = True
    for j in range( 1000 ) :
        d = digits( make_int( d, 2 ) + make_int( d[::-1], 2 ), 2 )[1:]
        if d == d[::-1] :
            it.append( j + 1 )
            isLychrel = False
            break
    if isLychrel :
        it.append( 0 )
print('Maximum iterations for non-seed numbers', max( it ))
Lychrel = []
for i in range( len(it) ) :
    if it[i] == 0 :
        Lychrel.append( i + 1 )
print('Count of binary Lychrel numbers', len( Lychrel ))
print('All binary lichler under', maxn)
print('Decimal form', Lychrel)
print('Binary form', list(map( lambda x: ''.join( map( str, toSystem( x, 2 ) ) ), Lychrel )))

A277338 Reverse and Add! sequence starting with 295.

Original entry on oeis.org

295, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007, 8801197801088, 17602285712176, 84724043932847, 159547977975595

Offset: 0

Matt C. Anderson, Oct 09 2016

Apart from the initial term in both sequences, the same as A006960.
a(0) = 295; a(n+1) = a(n) + A004086(a(n)).
295 is conjectured to be the second smallest initial term which does not lead to a palindrome. Also, 196 is possibly the smallest initial term which does not lead to a palindrome. a(0) = 196 is described in A006960.

			a(0) = 295
a(1) = 295 + 592 = 887
a(2) = 887 + 788 = 1675
...

Wikipedia, Lychrel number

Cf. A004086.
Almost the same as A006960.
See index entries at A023108.

with(StringTools):
revnum := proc (n)
local a, b, c;
description "to REVerse the digits of a NUMber";
a := convert(n, string);
b := Reverse(a);
c := convert(b, decimal, 10)
end proc;
f := 0;
e := 295;
count := 0;
while f <> e do
e := e+f;
f := revnum(e);
count := count+1
end do;

a[1] = 295; a[n_] := a[n] = FromDigits@ Reverse@ IntegerDigits@ # + # &@ a[n - 1]; Array[a, 29] (* Michael De Vlieger, Oct 14 2016 *)

terms(n) = my(x=295, i=0); while(1, print1(x, ", "); x=x+eval(concat(Vecrev(Str(x)))); i++; if(i==n, break))
/* Print initial 30 terms as follows: */
terms(30) \\ Felix Fröhlich, Nov 15 2016

a(n) = A006960(n) for n >= 1.

a(n) = A243238(295, n+1). - Felix Fröhlich, Nov 20 2016

A320516 Palindromic wing primes that are also Lychrel candidates.

Original entry on oeis.org

7774777, 777767777, 77777677777, 99999199999, 1111118111111, 7777774777777, 111111181111111, 333333373333333, 77777777677777777, 99999999299999999, 9999999992999999999, 33333333333733333333333, 77777777777677777777777, 333333333333373333333333333

Offset: 1

Robert James Liguori, Oct 29 2018

Lychrel candidates are natural numbers that seem unable to form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers.

On January 23, 2017 a Russian schoolboy, Andrey S. Shchebetov, announced on his web site that he had found a sequence of the first 126 numbers (125 of them never reported before) that take exactly 261 steps to reach a 119-digit palindrome. That sequence was published in the OEIS as A281506. The trajectory of the last number of that sequence, 1186061987030929990, under the "Reverse and Add!" operation was published separately in the OEIS as A281507.

Robert Liguori, Github project SequencePier

Cf. A077798, A000040, A023108, A281506, A281507.

Seven terms inserted by Jon E. Schoenfield, Oct 31 2018

a(14) from Jon E. Schoenfield, Nov 01 2018

cp's OEIS Frontend

A090062 There is (presumably) one and only one palindrome in the Reverse and Add! trajectory of n.

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A090063 Numbers n such that there are (presumably) two palindromes in the Reverse and Add! trajectory of n.

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A090064 Numbers n such that there are (presumably) three palindromes in the Reverse and Add! trajectory of n.

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A090065 Numbers n such that there are (presumably) four palindromes in the Reverse and Add! trajectory of n.

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A090066 Numbers n such that there are (presumably) five palindromes in the Reverse and Add! trajectory of n.

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A090067 Numbers n such that there are (presumably) six palindromes in the Reverse and Add! trajectory of n.

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A090068 Numbers n such that there are (presumably) seven palindromes in the Reverse and Add! trajectory of n.

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A160078 Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits in binary expansion reversed). Binary analog of Lychrel numbers.

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Python

A277338 Reverse and Add! sequence starting with 295.

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