A075421 -id:A075421 - cp's OEIS frontend

A306596 Trajectory of n under the Reverse and Add! operation carried out in base 8 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

1021, 3623, 4327, 4382, 4404, 4413, 4444, 4500, 4502, 4518, 4522, 4528, 4530, 4575, 4592, 4594, 5117, 5502, 5629, 6270, 7806, 8247, 8607, 12503, 12527, 12535, 16319, 16383, 16815, 20711, 20975, 24751, 25015, 28351, 28415, 28671, 28775, 28791, 33757, 33766, 34254, 34286, 34757, 34781, 35268, 35276

Offset: 1

A.H.M. Smeets, Feb 27 2019

A number is considered here (presumably) a Lychrel number in base 8 if it does not reach a palindrome within 100 steps more than the actual record. For those record numbers of steps, see A306600; for the corresponding record-setting numbers, see A306599. Futhermore, a Lychrel number is considered not to reach the trajectory of any smaller Lychrel number if it does not reach a trajectory of a smaller Lychrel number within 100 steps more than the actual record. For those record number of steps see A306851, and its corresponding record setting numbers, see A306850.
For a(11) = 4522 we obtain a cyclic structure of the terms in its trajectory (starting at the 12th term in the trajectory) which can be represented by the context-free grammar with alphabet = {0,1,2,3,4,5,6,7} and production rules:
  S -> S_a | S_b | S_c | S_d | S_e | S_f | S_g | S_h,
  S_a -> 10 T_a  00, T_a -> 7 T_a 0 | 777670,
  S_b -> 11 T_b  01, T_b -> 0 T_b 7 | 076667,
  S_c -> 22 T_c  12, T_c -> 0 T_c 7 | 065557,
  S_d -> 44 T_d  34, T_d -> 0 T_d 7 | 043337,
  S_e -> 10 T_e 000, T_e -> 7 T_e 0 | 777670,
  S_f -> 11 T_f 701, T_f -> 0 T_f 7 | 007567,
  S_g -> 22 T_g 712, T_g -> 0 T_g 7 | 006357,
  S_h -> 44 T_h 734, T_h -> 0 T_h 7 | 003737;
i.e., the cycle length is 8.
For all other terms up to and including a(649) = 527823, no such structure has been obtained.

Cf. A306600, A306599, A306851, A306850.

Base-8 analog of A075252 (base 2), A077405 (base 3), A075421 (base 4) and A063048 (base 10).

A076247 Trajectory of 1059774 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

1059774, 4187583, 8355006, 20822715, 83391660, 144328605, 268919295, 1339676160, 1349598705, 2683144950, 5361370860, 9358549725, 17380163775, 85563883200, 89574690225, 173801637750, 343262166720, 359352580785

Offset: 0

Klaus Brockhaus, Oct 03 2002

1059774 = A075421(1096) is the fourth term of A075421 whose base 4 trajectory provably does not contain a palindrome. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below. - The generating function given describes the sequence from a(16) onward; the g.f. for the complete sequence is known but nearly twice as big.

			1059774 (decimal) = 10002232332 -> 10002232332 + 23323220001 = 33332112333 = 4187583 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A075421, A075153, A075466, A075467, A076248.

NestWhileList[# + IntegerReverse[#, 4] &, 1059774,  # !=
IntegerReverse[#, 4] &, 1, 23] (* Robert Price, Oct 18 2019 *)

{m=1059774; stop=19; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0), ..., a(15) as above; for n > 15 and n = 4 (mod 6): a(n) = 5*4^(2*k+12)-5237765*4^k where k = (n+2)/6; n = 5 (mod 6): a(n) = 5*4^(2*k+12)+246174955*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 10*4^(2*k+12)+157132950*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 20*4^(2*k+12)-20951060*4^k where k = (n-1)/6; n = 2 (mod 6): a(n) = 20*4^(2*k+12)+230461660*4^k-15 where k = (n-2)/6; n = 3 (mod 6): a(n) = 40*4^(2*k+12)+125706360*4^k-10 where k = (n-3)/6. G.f.: -15*(185397326496*x^11+95559181296*x^10+91268404224*x^9-183251937960*x^8-92341098492*x^7 -91268404224*x^6-48628806952*x^5-27174921532*x^4-22884144448*x^3+46483418410*x^2 +23956838719*x+22884144448)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

A076248 Trajectory of 1059831 under the Reverse and Add! operation carried out in base 4, written in base 10.

Original entry on oeis.org

1059831, 4728312, 7831065, 14433270, 24913965, 56412450, 92165625, 208908750, 396926625, 710289750, 1336954560, 1398889905, 2715199350, 5363547840, 5614238385, 10894222710, 21453945600, 21701687025, 43073052150

Offset: 0

Klaus Brockhaus, Oct 03 2002

1059831 = A075421(1105 ) is the fifth term of A075421 whose base 4 trajectory provably does not contain a palindrome. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below.

			1059831 (decimal) = 10002233313 -> 10002233313 + 31333220001 = 102002113320 = 4728312 (decimal).

Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A075421, A075153, A075466, A075467, A076247.

NestWhileList[# + IntegerReverse[#, 4] &, 1059831,  # != IntegerReverse[ #, 4] &, 1, 23] (* Robert Price, Oct 19 2019 *)

{m=1059831; stop=19; c=0; while(c0,d=divrem(k,4); k=d[1]; rev=4*rev+d[2]); c++; m=m+rev)}

a(0), ..., a(7) as above; for n > 7 and n = 2 (mod 6): a(n) = 5*4^(2*k+9)+3836395*4^k-15 where k = (n+4)/6; n = 3 (mod 6): a(n) = 10*4^(2*k+9)+2450070*4^k-10 where k = (n+3)/6; n = 4 (mod 6): a(n) = 20*4^(2*k+9)-326420*4^k where k = (n+2)/6; n = 5 (mod 6): a(n) = 20*4^(2*k+9)+3544540*4^k-15 where k = (n+1)/6; n = 0 (mod 6): a(n) = 40*4^(2*k+9)+1927800*4^k-10 where k = n/6; n = 1 (mod 6): a(n) = 80*4^(2*k+9)-322580*4^k where k = (n-1)/6. G.f.: -3*(668508000*x^19+444361200*x^18+222142800*x^17-528080680*x^16-356464620*x^15 -125753060*x^14-299532884*x^13-188180432*x^12-143040640*x^11+128992350*x^10+90219415*x^9 +38288125*x^8+28112975*x^7+6666425*x^6+5752375*x^5+424135*x^4+3044705*x^3+2610355*x^2 + 1576104*x+353277)/((x-1)*(x^2+x+1)*(2*x^3-1)*(2*x^3+1)*(4*x^3-1))

A091677 a(n) = smallest non-palindromic k such that the base-4 Reverse and Add! trajectory of k is palindrome-free and joins the trajectory of A091675(n).

Original entry on oeis.org

469892287, 318, 68346, 66349, 269237759, 272353, 110333, 1082314, 4279, 3903, 1049659, 290, 1210, 4334, 275436, 4199, 73784, 2082046, 5046, 4212653, 1052467, 4768988414, 1073998008, 1051069, 1058784, 719, 795, 799, 265038, 119810013

Offset: 1

Klaus Brockhaus, Jan 28 2004

a(1), a(5), a(22), a(23) and a(30) are conjectural; it is not yet ensured that they are minimal.
a(n) >= A091675(n); a(n) = A091675(n) iff the trajectory of A091675(n) is palindrome-free, i.e., A091675(n) is also a term of A075421.
a(n) determines a 1-1-mapping from the terms of A091675 to the terms of A075421, the inverse of the mapping determined by A091676.
The 1-1 property of the mapping depends on the conjecture that the base-4 Reverse and Add! trajectory of each term of A091675 contains only a finite number of palindromes (cf. A091680).
Base-4 analog of A089494.

			A091675(2) = 3, the trajectory of 3 joins the trajectory of 318 = A075421(2) at 20966400, so a(2) = 318. A091675(4) = 22, the trajectory of 22 joins the trajectory of 66349 = A075421(130) at 600785, so a(4) = 66349.

Index entries for sequences related to Reverse and Add!

Cf. A075421, A091675, A091676, A091680, A089494.

A091680 Smallest number whose base-4 Reverse and Add! trajectory (presumably) contains exactly n base-4 palindromes, or -1 if there is no such number.

Original entry on oeis.org

290, 78, 18, 6, 3, 36, 21, 19, 7, 8, 4, 2, 1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Klaus Brockhaus, Jan 28 2004

Conjecture 1: For each k > 0 the trajectory of k eventually leads to a term in the trajectory of some j which belongs to A075421, i.e., whose trajectory (presumably) never leads to a palindrome. Conjecture 2: There is no k > 0 such that the trajectory of k contains more than twelve palindromes, i.e., a(n) = -1 for n > 12.

Base-4 analog of A077594.

			a(4) = 3 since the trajectory of 3 contains the four palindromes 3, 15, 975, 64575 (3, 33, 33033, 3330333 in base 4) and at 20966400 joins the trajectory of 318 = A075421(2) and the trajectories of 1 (A035524) and 2 do not contain exactly four palindromes.

Index entries for sequences related to Reverse and Add!

Cf. A075299, A035524, A014192, A075420, A075421, A077594.

A344119 Numbers k whose trajectory under the Reverse and Add! operation carried out in base 16 does not reach a palindrome and (presumably) does not join the trajectory of any term m < k.

Original entry on oeis.org

413, 429, 443, 445, 3407, 3647, 3711, 3775, 3807, 3839, 4287, 7417, 12463, 12527, 16383, 24575, 28879, 45183, 45231, 49151, 57343, 61615, 61663, 61679, 66511, 66783, 67023, 67535, 67551, 67628, 67630, 67644, 67646, 67673, 67674, 67676, 67688, 67718, 67734

Offset: 1

A.H.M. Smeets, Aug 16 2021

A number is considered here (presumably) a Lychrel number in base 16 if it does not reach a palindrome within 200 steps more than the actual record. Those record numbers of steps to become palindromic are known from data in other bases not to increase that much (see for instance A065198 and A065199 in case of base 10). Furthermore, a Lychrel number is considered not to reach the trajectory of any smaller Lychrel number if it does not reach a trajectory of a smaller Lychrel number within 100 steps more than the actual record. Again, those record numbers of steps to reach the trajectory of a smaller Lychrel number are known from data in other bases not to increase that much (see for instance A323975 and A323976 in case of base 10).

In other bases: A075252 (2), A077405 (3), A075421 (4), A306596 (8), A063048 (10).

Cf. A065198, A065199, A323975, A323976.

cp's OEIS Frontend

A306596 Trajectory of n under the Reverse and Add! operation carried out in base 8 (presumably) does not reach a palindrome and (presumably) does not join the trajectory of any term m < n.

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A076247 Trajectory of 1059774 under the Reverse and Add! operation carried out in base 4, written in base 10.

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A076248 Trajectory of 1059831 under the Reverse and Add! operation carried out in base 4, written in base 10.

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A091677 a(n) = smallest non-palindromic k such that the base-4 Reverse and Add! trajectory of k is palindrome-free and joins the trajectory of A091675(n).

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A091680 Smallest number whose base-4 Reverse and Add! trajectory (presumably) contains exactly n base-4 palindromes, or -1 if there is no such number.

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A344119 Numbers k whose trajectory under the Reverse and Add! operation carried out in base 16 does not reach a palindrome and (presumably) does not join the trajectory of any term m < k.

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