A023108 -id:A023108 - cp's OEIS frontend

A063065 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063064 in order to obtain a term in the trajectory of 10563.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 1, 0

Offset: 0

Klaus Brockhaus, Jul 07 2001

			12543 is a term of A063064. One 'Reverse and Add!' operation applied to 12543 leads to a term (47064) in the trajectory of 10563, so the corresponding term of the present sequence is 1.

Index entries for sequences related to Reverse and Add!

A023108, A033865, A063063, A063064.

A063434 Integers k > 10577 such that the 'Reverse and Add!' trajectory of k joins the trajectory of 10577.

Original entry on oeis.org

11567, 12557, 13547, 14537, 15527, 16517, 17507, 20576, 21566, 22556, 23546, 24536, 25526, 26516, 27506, 30575, 31565, 32555, 33545, 34535, 35525, 36515, 37505, 40574, 41564, 42554, 43544, 44534, 45524, 46514, 47504, 50573, 51563

Offset: 0

Klaus Brockhaus, Jul 20 2001

Subsequence of A023108.

The first term not congruent 83 mod 99 is a(47) = 70069, thereafter the residues show no obvious pattern. - Klaus Brockhaus, Jul 14 2003

			The trajectory of 12557 reaches 88078 in one step and 88078 is a term in the trajectory of 10577, so 12557 belongs to the present sequence. The corresponding term in A063435, giving the number of steps, accordingly is 1.

Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A063433, A063435.

A066055 Integers n > 10583 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 10583.

Original entry on oeis.org

11573, 12563, 13553, 13597, 14543, 14587, 15533, 15577, 16523, 16567, 17513, 17557, 18097, 18503, 18547, 19087, 19537, 20582, 21572, 22562, 23552, 23596, 24586, 25532, 25576, 26522, 26566, 27512, 27556, 28096, 28502, 28546, 29086

Offset: 0

Klaus Brockhaus, Nov 30 2001

Subsequence of A023108.

			The trajectory of 13597 reaches 937838 in three steps and 937838 is a term in the trajectory of 10583, so 13597 belongs to the present sequence. The corresponding term in A066056, giving the number of steps, accordingly is 3.

Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A066054, A066056.

A090069 Numbers n such that there are (presumably) eight palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

3, 8, 20, 22, 100, 101, 116, 122, 139, 151, 160, 215, 221, 238, 313, 314, 320, 337, 343, 413, 436, 512, 611, 634, 696, 710, 717, 727, 733, 832, 931, 1004, 1011, 1070, 1101, 1160, 1250, 1340, 1430, 1520, 1610, 1700, 1771, 2000, 2002, 2003, 2010, 2100, 2112

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms <= 5000 each palindrome is reached from the preceding one or from the start in at most 15 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 8 begins 8, 16, 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 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the eight palindromes in the trajectory of 8 and 8 is a term.

Index entries for sequences related to Reverse and Add!

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

A090070 Numbers n such that there are (presumably) nine palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

4, 10, 11, 535, 1000, 1001, 10007, 10101, 20006, 30005, 50003, 60002, 70001, 80000, 80008, 100070, 110060, 120050, 130040, 140030, 150020, 160010, 170000, 170071, 200000, 200002, 1000003, 1000150, 1001001, 1010050, 1100140, 1110040, 1200130

Offset: 1

Klaus Brockhaus, Nov 20 2003

For terms < 5000000 each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 2000 steps.

			The trajectory of 4 begins 4, 8, 16, 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 4, 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the nine palindromes in the trajectory of 4 and 4 is a term.

Index entries for sequences related to Reverse and Add!

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

A090071 Numbers n such that there are (presumably) ten palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

2, 5, 10003, 30001, 40000, 40004, 100000, 100001, 2000000, 2000002

Offset: 1

Klaus Brockhaus, Nov 20 2003

Additional terms are 20000000, 20000002, 200000000, 200000002, 2000000000, 2000000002, 10000000004, 10000100001, 20000000000, 20000000002, 20000000003, 30000000002, 40000000001, but it is not yet ascertained that they are consecutive.

For all terms given above each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 5000 steps.

			The trajectory of 2 begins 2, 4, 8, 16, 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 2, 4, 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the ten palindromes in the trajectory of 2 and 2 is a term.

Index entries for sequences related to Reverse and Add!

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

A090072 Numbers n such that there are (presumably) eleven palindromes in the Reverse and Add! trajectory of n.

Original entry on oeis.org

1, 20000, 20002, 1000000, 1000001, 10000000, 10000001

Offset: 1

Klaus Brockhaus, Nov 20 2003

Additional terms (cf. A090075) are 100000000, 100000001, 100010001, 1000000000, 1000000001, 10000000000, 10000000001, 100000000000, 100000000001, 1000000000000, 1000000000001, 1000001000001, 1000100010001, but it is not yet ascertained that they are consecutive.
For all terms given above each palindrome is reached from the preceding one or from the start in at most 35 steps; after the presumably last one no further palindrome is reached in 5000 steps.
Only two numbers are known whose Reverse and Add trajectory contains twelve palindromes: 10000 and 10001. It is conjectured that these are the only such numbers and it has been conjectured before (cf. A077594) that no Reverse and Add trajectory contains more than twelve palindromes.

			The trajectory of 1 begins 1, 2, 4, 8, 16, 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 1, 2, 4, 8, 77, 1111, 2222, 4444, 8888, 661166 and 3654563 are the eleven palindromes in the trajectory of 1 and 1 is a term.

Index entries for sequences related to Reverse and Add!

Cf. A023108, A023109, A065001, A070742, A077594, A090075.

A247128 Positive numbers that are congruent to {0,5,9,13,17} mod 22.

Original entry on oeis.org

5, 9, 13, 17, 22, 27, 31, 35, 39, 44, 49, 53, 57, 61, 66, 71, 75, 79, 83, 88, 93, 97, 101, 105, 110, 115, 119, 123, 127, 132, 137, 141, 145, 149, 154, 159, 163, 167, 171, 176, 181, 185, 189, 193, 198, 203, 207, 211

Offset: 1

Karl V. Keller, Jr., Nov 19 2014

This sequence is the union of 22*n-17, 22*n-13, 22*n-9, and 22*n-5, and A008604(22*n), for n>0.
This sequence is the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; see example.
The sequence numbers with both odd first and last digits are either palindromes or they have corresponding reversed digit numbers, e.g., 105, 501. Prime numbers in this sequence are also in A007500 (reversal primes). Some examples are 13, 17, 31, 71, 79, 97, 101.
The sequence numbers with even first digits and last digits of 2, 4, 6 or 8, are either palindromes or they have corresponding reversed digit numbers in this sequence.
The candidate Lychrel numbers, 295, 493, 691, 1677, 1765, 1857, 1945, 1997, 3493, are in this sequence.

			Sequence consists of the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; e.g.,
for k =  5, sqrt( 20 -  2 + 3 + 1) = sqrt(22)  =  4.6904;
for k =  6, sqrt( 24 -  2 + 3 + 0) = sqrt(25)  =  5;
for k = 21, sqrt( 84 -  7 + 3 + 1) = sqrt(81)  =  9;
for k = 44, sqrt(176 - 15 + 3 + 0) = sqrt(164) = 12.8062;
for k = 45, sqrt(180 - 15 + 3 + 1) = sqrt(169) = 13.
Of these, the only integer values are 5, 9, 13, so they are in the sequence.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, 196 Algorithm.
Eric Weisstein's World of Mathematics, Lychrel Number
Eric Weisstein's World of Mathematics, Palindromic Number Conjecture
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A008604, A002113 (palindromes), A007500 (reversible primes).

Cf. A023108.

a247128[n_Integer] := Select[Range[n], MemberQ[{0, 5, 9, 13, 17}, Mod[#, 22]] &]; a247128[211] (* Michael De Vlieger, Nov 23 2014 *)

isok(n) = m = n % 22; (m==0) || (m==5) || (m==9) || (m==13) || (m==17);
select(x->isok(x), vector(200, i, i)) \\ Michel Marcus, Nov 28 2014

from math import *
for n in range(0,100001):
  if (sqrt(4*n-ceil(n/3)+3+n%2))%1==0:print(int(sqrt(4*n-ceil(n/3)+3+n%2)),end=",")

A247128_list = [n for n in range(1,10**5) if (n % 22) in {0,5,9,13,17}]
# Chai Wah Wu, Dec 31 2014

A247128_list, l = [], [5,9,13,17,22]
for _ in range(10**5):
    A247128_list.extend(l)
    l = [x+22 for x in l] # Chai Wah Wu, Jan 01 2015

a(n) = a(n-1) + a(n-5) - a(n-6). - Colin Barker, Nov 20 2014
G.f.: x*(5*x^4+4*x^3+4*x^2+4*x+5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Nov 20 2014
Proof that a(n) = a(n-1) + a(n-5) - a(n-6): the sequence a(n) is a concatenation of the sequences [5+22*i, 9+22*i, 13+22*i, 17+22*i, 22+22*i] for i = 0,1,2,..., so it is clear that a(n-1) = a(n-6) + 22 and a(n) = a(n-5) + 22. - Chai Wah Wu, Jan 01 2015

A063050 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063049 in order to obtain a term in the trajectory of 196.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1

Offset: 0

Klaus Brockhaus, Jul 07 2001

			394 is a term of A063049. One 'Reverse and Add!' operation applied to 394 leads to a term (887) in the trajectory of 196, so the corresponding term of the present sequence is 1.

Index entries for sequences related to Reverse and Add!

A023108, A033865, A006960, A063049.

limit = 10^3; x = NestList[ # + IntegerReverse[#] &, 196, limit];
y = Select[Range[197, 4942],
   Intersection[NestList[ # + IntegerReverse[#] &, #, limit],
      x] != {} &];
Table[
 Length@NestWhileList[# + IntegerReverse[#] &,
    y[[i]], ! MemberQ[x, #] &] - 1, {i, Length[y]}]
(* Robert Price, Oct 21 2019 *)

A063053 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063052 in order to obtain a term in the trajectory of 879.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 4, 5, 4, 3, 4, 5, 3, 4, 3, 4, 3, 4, 3, 1, 3, 0, 3, 1, 3, 1, 4, 5, 4, 5, 3, 4, 5, 3, 4, 3, 4, 3, 4, 3

Offset: 0

Klaus Brockhaus, Jul 07 2001

			1497 is a term of A063052. One 'Reverse and Add!' operation applied to 1497 leads to a term (9438) in the trajectory of 879, so the corresponding term of the present sequence is 1.

Index entries for sequences related to Reverse and Add!

A023108, A033865, A063051, A063052.

cp's OEIS Frontend

A063065 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063064 in order to obtain a term in the trajectory of 10563.

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A063434 Integers k > 10577 such that the 'Reverse and Add!' trajectory of k joins the trajectory of 10577.

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A066055 Integers n > 10583 such that the 'Reverse and Add!' trajectory of n joins the trajectory of 10583.

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A090069 Numbers n such that there are (presumably) eight palindromes in the Reverse and Add! trajectory of n.

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A090070 Numbers n such that there are (presumably) nine palindromes in the Reverse and Add! trajectory of n.

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A090071 Numbers n such that there are (presumably) ten palindromes in the Reverse and Add! trajectory of n.

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A090072 Numbers n such that there are (presumably) eleven palindromes in the Reverse and Add! trajectory of n.

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A247128 Positive numbers that are congruent to {0,5,9,13,17} mod 22.

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Formula

A063050 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063049 in order to obtain a term in the trajectory of 196.

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