A067030 -id:A067030 - cp's OEIS frontend

A067032 Number of k's such that A067030(n) = k + reverse(k).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 9, 9, 7, 1, 6, 5, 1, 4, 3, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 8, 9, 8

Offset: 0

Klaus Brockhaus, Dec 29 2001

			a(12) = 3 since A067030(12) = 33 and for k = 12, 21, 30 we have 33 = k + reverse(k).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A033865, A067030, A067031, A067033, A067034.

function a067032(a,b: integer); var n,k,c,i,rev: integer; st,nst: string; begin for n := a to b do k := 0; c := 0; while k <= n do st := itoa(k); nst := ""; for i := 0 to length(st) - 1 do nst := concat(st[i],nst); end; rev := atoi(nst); if n = k + rev then inc(c); end; inc(k); end; if c > 0 then write(c,","); end; end; end; a067032(0,1000);

A067034 Largest k such that A067030(n) = k + reverse(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 7, 8, 9, 20, 30, 40, 50, 60, 70, 80, 90, 100, 91, 110, 93, 120, 94, 95, 130, 96, 97, 140, 98, 99, 150, 200, 160, 210, 170, 220, 180, 230, 190, 240, 250, 300, 260, 310, 270, 320, 280, 330, 290, 340, 350, 400, 360, 410, 370, 420, 380, 430

Offset: 0

Klaus Brockhaus, Dec 29 2001

			a(12) = 30 since A067030(12) = 33 and 30 is the largest k such that 33 = k + reverse(k).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A033865, A067031, A067031, A067032, A067033.

function a067034(a,b: integer); var n,k,m,i,rev: integer; st,nst: string; begin for n := a to b do k := 0; m := -1; while k <= n do st := itoa(k); nst := ""; for i := 0 to length(st) - 1 do nst := concat(st[i],nst); end; rev := atoi(nst); if n = k + rev then m := k; end; inc(k); end; if m >= 0 then write(m,","); end; end; end; a067034(0,500);

A067033 Smallest k such that A067030(n) = k + reverse(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 100, 19, 29, 39, 120, 49, 59, 130, 69, 79, 140, 89, 99, 150, 101, 160, 111, 170, 121, 180, 131, 190, 141, 151, 102, 161, 112, 171, 122, 181, 132, 191, 142, 152, 103, 162, 113, 172, 123, 182, 133, 192

Offset: 0

Klaus Brockhaus, Dec 29 2001

			a(12) = 12 since A067030(12) = 33 and 12 is the smallest k such that 33 = k + reverse(k).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A033865, A067030, A067031, A067032, A067034.

function a067033(a,b: integer); var n,k,i,rev: integer; st,nst: string; begin for n := a to b do k := 0; while k <= n do st := itoa(k); nst := ""; for i := 0 to length(st) - 1 do nst := concat(st[i],nst); end; rev := atoi(nst); if n = k + rev then write(k,",") k := n + 1; else inc(k); end; end; end; end; a067033(0,500);

A067284 a(n) = number of integers k such that k is not of the form m + reverse(m) for any m (cf. A067031) and A067030(n) occurs in the 'Reverse and Add' trajectory of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 5, 8, 7, 12, 9, 1, 13, 21, 14, 1, 6, 11, 1, 4, 14, 1, 2, 9, 1, 2, 1, 2, 1, 22, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 16, 2, 3, 3, 5, 3, 4, 3, 5, 3, 4, 3, 30, 4, 6, 5, 5, 4, 6, 5, 5, 4, 6, 15, 8, 6, 6, 5, 8, 6, 6, 5, 8, 6, 9, 24, 7, 6, 9, 8, 7, 6, 9, 7, 12, 9, 8, 20

Offset: 0

Klaus Brockhaus, Feb 04 2002

For an integer j not in A067030 there exists no integer k of the form m + reverse(m) such that j occurs in the trajectory of k. a(n) >= A067032(n); A067737 gives the terms of A067030 such that a(n) > A067032(n). A067288 gives the records in this sequence, A067287 gives the terms of A067030 at which these records are attained.

			a(14) = 5, since A067030(14) = 55 and the five integers 7, 23, 32, 41, 50 are not of the form m + reverse(m) for any m and 55 occurs in the trajectory of each of them. a(25) = 11, since A067030(25) = 154 and the eleven integers 1, 25, 34, 43, 52, 59, 61, 68, 70, 86, 95 are not of the form m + reverse(m) for any m and 154 occurs in the trajectory of each of them.

Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067032, A067287, A067288, A067737.

A067286 a(n) = largest integer k such that k is not of the form m + reverse(m) for any m (cf. A067031) and A067030(n) occurs in the 'Reverse and Add' trajectory of k.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 5, 3, 7, 1, 9, 20, 30, 40, 50, 60, 70, 80, 90, 100, 91, 92, 93, 120, 94, 95, 130, 96, 97, 140, 98, 90, 150, 200, 160, 210, 170, 220, 180, 230, 190, 240, 250, 300, 260, 310, 270, 320, 280, 330, 290, 340, 350, 400, 360, 410, 370, 420, 380, 430, 390

Offset: 0

Klaus Brockhaus, Feb 04 2002

a(n) <= A067034(n). If A067034(n) is in A067030 then a(n) < A067034(n), otherwise a(n) = A067034(n).

			a(14) = 50, since A067030(14) = 55 and the five integers 7, 23, 32, 41, 50 are not of the form m + reverse(m) for any m, 55 occurs in the trajectory of each of them and 50 is the largest one. a(25) = 95, since A067030(25) = 154 and the eleven integers 1, 25, 34, 43, 52, 59, 61, 68, 70, 86, 95 are not of the form m + reverse(m) for any m, 154 occurs in the trajectory of each of them and 95 is the largest one.

Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067034, A067284, A067285.

A067288 Records for the number of integers k such that k is not of the form m + reverse(m) for any m and for some n A067030(n) occurs in the 'Reverse and Add' trajectory of k (cf. A067284).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 13, 21, 22, 30, 38, 39, 42, 46, 71, 90, 94, 150, 254, 286, 404, 434, 578, 586, 602, 643, 758, 799, 813, 847, 1131, 1162, 1169, 1334, 1742, 2093, 2120, 2378, 2663, 2892, 3208, 3383, 3585, 3685, 3999, 4818, 4942, 5766, 5959

Offset: 1

Klaus Brockhaus, Feb 04 2002

Successive maxima in sequence A067284. A067287 gives the corresponding integers at which these records are attained.

			3 is a record, since for A067030(12) = 33 there are three integers k not of the form j + reverse(j) for any j such that 33 occurs in the "Reverse and Add!" trajectory of these k and for m < 33 there are at most two integers which have the corresponding property.

Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067032, A067035, A067036, A067284, A067287.

More terms from Larry Reeves (larryr(AT)acm.org), Dec 18 2002

Offset and a(27) onward corrected by Sean A. Irvine, Dec 12 2023

A067285 a(n) = smallest integer k such that k is not of the form m + reverse(m) for any m (cf. A067031) and A067030(n) occurs in the 'Reverse and Add' trajectory of k.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 5, 3, 7, 1, 9, 5, 3, 5, 7, 3, 1, 5, 9, 100, 7, 7, 3, 120, 49, 1, 130, 69, 5, 140, 89, 9, 150, 100, 160, 111, 170, 7, 180, 131, 190, 120, 151, 102, 130, 112, 171, 122, 140, 3, 191, 142, 152, 100, 162, 113, 172, 111, 182, 133, 192, 7, 153, 104, 131, 114

Offset: 0

Klaus Brockhaus, Feb 04 2002

a(n) <= A067033(n).

			a(14) = 7, since A067030(14) = 55 and the five integers 7, 23, 32, 41, 50 are not of the form m + reverse(m) for any m, 55 occurs in the trajectory of each of them and 7 is the smallest one. a(25) = 1, since A067030(25) = 154 and the eleven integers 1, 25, 34, 43, 52, 59, 61, 68, 70, 86, 95 are not of the form m + reverse(m) for any m, 154 occurs in the trajectory of each of them and 1 is the smallest one.

Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067033, A067284, A067286.

A068798 Integers n such that n = A067030(j) for some j and A067286(j) < A067034(j).

Original entry on oeis.org

4, 8, 11, 12, 16, 121, 198, 1717, 1757, 1797, 1818, 1837, 1858, 1877, 1898, 1938, 1978, 11011, 17127, 18018, 18887, 19998, 111001, 113201, 115401, 117601, 119801, 170217, 170617, 171017, 171227, 171417, 171627, 171817, 172027, 172427, 172827, 180018, 180418

Offset: 1

Klaus Brockhaus, Mar 05 2002

Integers n such that n = A067030(j) for some j and [largest integer k such that k is not of the form m + reverse(m) for any m (cf. A067031) and n occurs in the 'Reverse and Add' trajectory of k.] is smaller than [largest k such that n = k + reverse(k)]. - A067030(j) is a term iff A067034(j) is in A067030.

			4 = A067030(2) is in the sequence, since A067286(2) = 1 < 2 = A067034(2). 121 = A067030(21) is in the sequence, since A067286(21) = 92 < 110 = A067034(21).

Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067034, A067286.

More terms from Sean A. Irvine, Mar 15 2024

A358513 a(n) is the smallest number whose divisors include exactly n that can be written in the form m + reverse(m), for some m (A067030).

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 48, 88, 220, 176, 132, 968, 264, 396, 528, 792, 1320, 1584, 2640, 3960, 5808, 5544, 8712, 14520, 11088, 24024, 21780, 36036, 40656, 39996, 53328, 87120, 60984, 72072, 205128, 132132, 121968, 144144, 293304, 199980, 266640, 439956, 264264, 360360, 733260, 396396, 660660, 799920

Offset: 0

Marius A. Burtea, Dec 04 2022

			1 has no divisors that can be written in the form m + reverse(m), so a(0) = 1.
2 has only the divisor 2 which is written 2 = 1 + reverse(1), so a(1) = 2.
3 has no divisors that can be written in the form m + reverse(m).
4 has divisors 1, 2, 4 but only 2 = 1 + reverse(1) and 4 = 2 + reverse(2), so a(2) = 4.
5 and 7 have no divisors that can be written in the form m + reverse(m), and 6 only has the divisors 2 = 2 + reverse(2) and 6 = 3 + reverse(3).
8 has divisors 1, 2, 4, 8 but only 2 = 1 + reverse(1), 4 = 2 + reverse(2) and 8 = 4 + reverse(4), so a(3) = 8.

Cf. A067032, A067030.

rev:=func; f:=func; a:=[]; for n in [0..25] do k:=1; while #[d:d in Divisors(k)|f(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

rev:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc:
S:= select(`<=`, map(t -> t + rev(t), {$1..10^6}),10^6):
V:= Array(0..49): count:= 0:
for n from 1 to 10^6 while count < 50 do
  v:= nops(numtheory:-divisors(n) intersect S);
  if v <= 49 and V[v] = 0 then
     count:= count+1; V[v]:= n;
  fi
od:
convert(V,list); # Robert Israel, Dec 28 2022

More terms from Robert Israel, Dec 28 2022

A056964 a(n) = n + reversal of digits of n.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 66, 77, 88, 99, 110

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits then a(n) is a multiple of 11.

Also called the Reverse and Add!, or RADD operation. Iteration of this function leads to the definition of Lychrel and related numbers, cf. A023108, A063048, A088753, A006960, and many others. - M. F. Hasler, Apr 13 2019

			a(17) = 17 + 71 = 88.

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (first 1001 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Reverse-Then-Add Sequence
Index entries for sequences related to Reverse and Add!

Cf. A004086, A056965, A067030.
Differs from A052008 when n=101 and a(101)=202 while A052008(101)=121
Cf. A036839.

a056964 n = n + a004086 n  -- Reinhard Zumkeller, Oct 14 2011

Table[n+FromDigits[Reverse[IntegerDigits[n]]],{n,0,100}] (* Harvey P. Dale, Jul 19 2014 *)
#+IntegerReverse[#]&/@Range[0,100] (* Harvey P. Dale, Jul 31 2025 *)

A056964(n)=fromdigits(Vecrev(digits(n)))+n \\ Charles R Greathouse IV, Oct 28 2014

def A056964(n): return n+int(str(n)[::-1]) # Indranil Ghosh, Jan 29 2017

a(n) = n + A004086(n) = 2*n - A056965(n).

n < a(n) < 11n for n > 0. - Charles R Greathouse IV, Nov 17 2022

cp's OEIS Frontend

A067032 Number of k's such that A067030(n) = k + reverse(k).

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A067034 Largest k such that A067030(n) = k + reverse(k).

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A067033 Smallest k such that A067030(n) = k + reverse(k).

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A067284 a(n) = number of integers k such that k is not of the form m + reverse(m) for any m (cf. A067031) and A067030(n) occurs in the 'Reverse and Add' trajectory of k.

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A067286 a(n) = largest integer k such that k is not of the form m + reverse(m) for any m (cf. A067031) and A067030(n) occurs in the 'Reverse and Add' trajectory of k.

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A067288 Records for the number of integers k such that k is not of the form m + reverse(m) for any m and for some n A067030(n) occurs in the 'Reverse and Add' trajectory of k (cf. A067284).

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A067285 a(n) = smallest integer k such that k is not of the form m + reverse(m) for any m (cf. A067031) and A067030(n) occurs in the 'Reverse and Add' trajectory of k.

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A068798 Integers n such that n = A067030(j) for some j and A067286(j) < A067034(j).

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A358513 a(n) is the smallest number whose divisors include exactly n that can be written in the form m + reverse(m), for some m (A067030).

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