A006960 -id:A006960 - cp's OEIS frontend

A063054 'Reverse and Add!' trajectory of 1997.

1997, 9988, 18887, 97768, 184547, 930028, 1750067, 9350638, 17711177, 94822948, 179745797, 977293768, 1844686547, 9301551028, 17503102067, 93523232638, 177146465177, 948711106948, 1798312224797, 9772534363768, 18446168716547

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 1997 + 7991 = 9988.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2318 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063055, A056964, A004086.

m := 1997; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063054 n = a063054_list !! n
a063054_list = iterate a056964 1997 -- Reinhard Zumkeller, Sep 22 2011

NestList[ #+FromDigits[ Reverse[ IntegerDigits[ # ] ] ]&, 1997, 25 ]
NestList[#+IntegerReverse[#]&,1997,25] (* Harvey P. Dale, Jul 15 2025 *)

A063060 'Reverse and Add!' trajectory of 10553.

Original entry on oeis.org

10553, 46054, 91118, 172237, 904508, 1709917, 8908988, 17807086, 85877957, 161855815, 680413976, 1359728062, 3968007593, 7925016286, 14751121583, 53263237324, 95636473559, 191173937218, 1003913308409, 10051946501410

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 10553 + 35501 = 46054.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2423 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063061.

Cf. A056964, A004086.

m := 10553; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063060 n = a063060_list !! n
a063060_list = iterate a056964 10553 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 10553, 40] (* Vincenzo Librandi, May 03 2014 *)

Updated b-file from Michael Lee, Apr 01 2012

A063063 'Reverse and Add!' trajectory of 10563.

Original entry on oeis.org

10563, 47064, 93138, 176277, 948948, 1798797, 9777768, 18455547, 93011028, 175022067, 935242638, 1771485177, 9487326948, 17983564797, 97730103768, 184460207547, 930162272028, 1750434533067, 9353788873638, 17717577747177

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 10563 + 36501 = 47064.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2383 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063064.

Cf. A056964, A004086.

m := 10563; stop := 25; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a063063 n = a063063_list !! n
a063063_list = iterate a056964 10563 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 10563, 30] (* Vincenzo Librandi, May 03 2014 *)

A066054 'Reverse and Add!' trajectory of 10583.

Original entry on oeis.org

10583, 49084, 97178, 184357, 937838, 1776577, 9533348, 17966707, 88733678, 176367466, 841131137, 1572262285, 7394885036, 13700769973, 51697470704, 92404950319, 183710890748, 1030808908129, 10248906988430, 13737867972631

Offset: 0

Klaus Brockhaus, Nov 30 2001

			a(1) = 10583 + 38501 = 49084.

Harry J. Smith and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (the first 150 terms from Harry J. Smith)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A066055.

Cf. A056964, A004086.

: m := 10583; stop := 20; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a066054 n = a066054_list !! n
a066054_list = iterate a056964 10583 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&,  10583, 40] (* Vincenzo Librandi, May 03 2014 *)

Rev(x)= { local(d, r=0); while (x>0, d=x%10; x\=10; r=r*10 + d); return(r) } { a=10583; for (n = 0, 150, if (n, a+=Rev(a)); write("b066054.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 08 2009

A244567 Triangle T(n,k) in which the n-th row lists in increasing order all values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s; triangle T(n,k), n>=0, 1<=k<=A244569(n), read by rows.

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 4, 5, 3, 6, 7, 1, 2, 4, 8, 9, 5, 10, 5, 10, 11, 3, 6, 12, 13, 7, 14, 15, 1, 2, 4, 8, 16, 17, 9, 18, 19, 20, 21, 5, 10, 11, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 6, 12, 21, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43

Offset: 0

Alois P. Heinz, Jun 30 2014

			Triangle T(n,k) begins:
   0;
   1;
   1,  2;
   3;
   1,  2,  4;
   5;
   3,  6;
   7;
   1,  2,  4,  8;
   9;
   5, 10;
   5, 10, 11;
   3,  6, 12;
  13;
   7, 14;
  15;
   1,  2,  4,  8,  16;
  17;

Alois P. Heinz, Rows n = 0..6000, flattened

Column k=0 gives A244568.
Last elements of rows give A001477.
Cf. A001127, A033648, A033670, A006960, A004086, A244569.

A244568 Smallest value s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 5, 3, 13, 7, 15, 1, 17, 9, 19, 20, 21, 5, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 5, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 7, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 3, 67, 68, 69

Offset: 0

Alois P. Heinz, Jun 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Column k=0 of A244567.

Cf. A001127, A033648, A033670, A006960, A004086, A244569.

A244569 Number of values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 3, 3, 1, 2, 1, 5, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17

Offset: 0

Alois P. Heinz, Jun 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Row lengths of A244567.

Cf. A001127, A033648, A033670, A006960, A004086, A244568.

A030547 Number of terms (including the initial term) needed to reach a palindrome when the Reverse Then Add! map (x -> x + (x-with-digits-reversed)) is repeatedly applied to n, or -1 if a palindrome is never reached.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 3, 2, 3, 3, 2, 2, 2, 2, 2, 1, 2, 3, 3, 4, 2, 2, 2, 2, 3, 2, 1, 3, 4, 5, 2, 2, 2, 3, 2, 3, 3, 1, 5, 7, 2, 2, 3, 2, 3, 3, 4, 5, 1, 25, 2, 3, 2, 3, 3, 4, 5, 7, 25

Offset: 1

Eric W. Weisstein

It is conjectured that a(196) is the smallest term equal to -1. See A023108.

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, 702 pages. See Entry 196.

Eric Weisstein's World of Mathematics, 196 Algorithm.

Cf. A006960, A023108, A063018, etc.

Equals A033665(n) + 1.

Table[Length@
  NestWhileList[# + IntegerReverse[#] &, n, ! PalindromeQ[#]  &], {n, 98}] (* Robert Price, Oct 18 2019 *)

Edited by N. J. A. Sloane, May 09 2015

A244058 n-th term of the 'Reverse and Add!' sequence starting with n.

Original entry on oeis.org

1, 4, 12, 77, 44, 363, 484, 4444, 79497, 14003, 88088, 175857, 1596595, 1716517, 17794887, 13528163, 176599676, 839546037, 1317544822, 853595347, 8836886388, 13236127322, 13297007933, 668823329856, 175304440267, 909153350908, 9317488848228, 8813200023188

Offset: 1

Alois P. Heinz, Jun 18 2014

a(n) is a palindrome for n in {1, 2, 4, 5, 6, 7, 8, 9, 11, 21, 28, 30}.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Reverse and Add!

Main diagonal of A243238.

Cf. A001127, A033648, A033670, A006960, A004086.

b:= proc(n, j) option remember; `if`(j=1, n, (h-> h+ (s->
      parse(cat(s[-i]$i=1..length(s))))(""||h))(b(n, j-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..40);

Table[Nest[#+IntegerReverse[#]&,n,n-1],{n,30}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 07 2021 *)

A273228 G.f. is the fourth power of the g.f. of A006950.

Original entry on oeis.org

1, 4, 10, 24, 55, 116, 230, 440, 819, 1480, 2602, 4480, 7580, 12604, 20620, 33272, 53029, 83520, 130088, 200600, 306488, 464168, 697150, 1039032, 1537435, 2259300, 3298428, 4785880, 6903657, 9903040, 14129846, 20058488, 28336790, 39845456, 55778050, 77747328, 107924347, 149221160

Offset: 0

M.S. Mahadeva Naika, May 18 2016

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.
L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.

Cf. A006960, A273226.

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q,2,1000)^4/etaq(q,1,1000)^4/etaq(q,4,1000)^4,q,50):
CoefficientList(sort(convert(GenFun,polynom),q,ascending),q);

nmax = 30; CoefficientList[Series[Product[(1 + x^k)^4 / (1 - x^(4*k))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 25 2017 *)
CoefficientList[Series[1/(QPochhammer[q, -q]*QPochhammer[q^2, q^2])^4, {q, 0, 50}], q] (* G. C. Greubel, Apr 17 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^4 / (1 - x^(4*k))^4, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.
a(n) ~ exp(sqrt(2*n)*Pi) / (2^(9/4)*n^(7/4)). - Vaclav Kotesovec, Mar 25 2017

Edited by N. J. A. Sloane, May 26 2016

cp's OEIS Frontend

A063054 'Reverse and Add!' trajectory of 1997.

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A063060 'Reverse and Add!' trajectory of 10553.

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A063063 'Reverse and Add!' trajectory of 10563.

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A066054 'Reverse and Add!' trajectory of 10583.

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A244567 Triangle T(n,k) in which the n-th row lists in increasing order all values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s; triangle T(n,k), n>=0, 1<=k<=A244569(n), read by rows.

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A244568 Smallest value s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

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A244569 Number of values s such that n is in the trajectory of the 'Reverse and Add!' sequence starting with s.

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A030547 Number of terms (including the initial term) needed to reach a palindrome when the Reverse Then Add! map (x -> x + (x-with-digits-reversed)) is repeatedly applied to n, or -1 if a palindrome is never reached.

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A244058 n-th term of the 'Reverse and Add!' sequence starting with n.

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