A067030 -id:A067030 - cp's OEIS frontend

A190481 Number of distinct integers with n digits which are the image of integers by the function Reverse and Add!.

4, 14, 93, 256, 1793, 4872, 34107, 92590, 648154, 1759313, 12315269, 33427272, 233991155, 635119194, 4445835138, 12067267861, 84470877438, 229278099157, 1604946701532, 4356283914175, 30493987422124, 82769394462323, 579385761306789, 1572618495070552

Offset: 1

Aldo González Lorenzo, May 25 2011

a(n) is the cardinality of the set of Image(Reverse and Add!) intersected with [10^(n-1), 10^n[. Here we suppose that the domain of the function Reverse and Add! is {1, 2, 3, ...}

There are 4, 50, 450, 4590, 45405,... (A232731) ways to obtain integers with n = 1,2,... digits as images under the function "Reverse and add!", but many result in the same image and are counted here only once. Example: 11+digrev(11) = 22 and 20+digrev(20)=22 contribute only once to the set of distinct images at n=2. - R. J. Mathar, Jun 17 2011

			Example: let RaA(x) be the function Reverse and Add!, then:
RaA(1)=2
RaA(2)=4
RaA(3)=6
RaA(4)=8
RaA(5)=10
RaA(6)=11, ...
So a(1) is the cardinal of {2,4,6,8}, which is 4:

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

Cf. A056964, A067030.

A055642 := proc(n) max(1,1+ilog10(n)) ; end proc:
A056964 := proc(n) n+digrev(n) ; end proc:
A190481 := proc(n) local s,i,ra ; s := {} ; for i from 1 to 10^n do ra := A056964(i) ; if A055642( ra) = n then s := s union {ra}  ; end if; end do: nops(s) ; end proc:
for n from 1 do print(n,A190481(n)) ; end do: # R. J. Mathar, Jun 17 2011

Empirical g.f.: x*(4 + 18*x + 23*x^2 - 29*x^3 - 58*x^4 - 34*x^5 - 81*x^6 - 45*x^7 - 32*x^8 - 9*x^9) / ((1 + x)*(1 - 19*x^2)*(1 - 2*x + x^2 - x^3)*(1 + 2*x + x^2 + x^3)). - Colin Barker, Mar 20 2017

a(9)-a(10) from Lars Blomberg, Dec 01 2013

a(11)-a(24) from Hiroaki Yamanouchi, Sep 04 2014

A227525 Squares which can be represented as sum of a prime and its reverse in at least one way.

Original entry on oeis.org

4, 121, 625, 94249, 698896, 1517824, 5313025, 6325225, 6895876, 6948496, 7706176, 15665764, 63600625, 95199049, 170198116, 449948944, 522808225, 562069264, 1101576100, 1183979281, 1254505561, 1271564281, 1615718416, 4045214404, 9504885049, 10989328900

Offset: 1

Shyam Sunder Gupta, Jul 14 2013

			121=29+92

Giovanni Resta, Table of n, a(n) for n = 1..56 (terms < 2*10^12)

Cf. A067030.

SquareQ[n_Integer?Positive] := IntegerQ[Sqrt[n]] ; Union[Select[ Table[ Prime[x] + FromDigits[ Reverse[ IntegerDigits[ Prime[x]]]], {x,  5761455}], SquareQ]]
Select[Union[#+IntegerReverse[#]&/@Prime[Range[10^6]]],IntegerQ[Sqrt[#]]&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Sep 14 2022 *)

a(20)-a(26) from Giovanni Resta, Jul 18 2013

A335978 Numbers m of the form abs(k - reverse(k)) for at least one k.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 180, 189, 198, 270, 279, 297, 360, 369, 396, 450, 459, 495, 540, 549, 594, 630, 639, 693, 720, 729, 792, 810, 819, 891, 900, 909, 990, 999, 1089, 1179, 1188, 1269, 1278, 1359, 1368, 1449, 1458, 1539, 1548, 1629, 1638, 1719, 1728, 1800, 1809, 1818, 1890, 1908, 1980, 1989, 1998, 2079

Offset: 1

Michael Greaney, Jul 03 2020

All terms are divisible by 9.
Let f(k) = k - reverse(k). Then f(reverse(k)) = -f(k), since f(reverse(k)) = reverse(k) - reverse(reverse(k)) = reverse(k) - k = - (k - reverse(k)) = -f(k).
Iteration of the function f(k) = k - reverse(k) leads to A072140, A072141, A072142, and A072143.

Michael P. Greaney, Remarkable Reversible Numbers, +plus magazine, (September 29, 2015).

Cf. A056965, A067030, A072140, A072141, A072142, A072143.

Dividing by 9 gives A334145.

A067737 Integers n such that [number of integers k such that k is not of the form m + reverse(m) for any m and n occurs in the "Reverse and Add!" trajectory of k] is greater than [number of integers k such that n = k + reverse(k)].

Original entry on oeis.org

44, 66, 88, 110, 121, 132, 154, 176, 198, 242, 363, 404, 444, 484, 505, 524, 545, 564, 585, 605, 606, 625, 646, 665, 686, 707, 726, 747, 766, 787, 808, 827, 847, 848, 867, 888, 909, 928, 949, 968, 989, 1010, 1029, 1050, 1069, 1089, 1090, 1111, 1130, 1151

Offset: 1

Klaus Brockhaus, Feb 04 2002

Integers n such that n = A067030(j) for some j and A067284(j) > A067032(j).

			44 = A067030(13) is in the sequence, since there are five integers k (viz. 5, 13, 20, 31, 40; A067284(13) = 5) such that k is not of the form m + reverse(m) for any m and 44 occurs in the "Reverse and Add!" trajectory of k, but only four integers k (viz. 13, 22, 31, 40; A067032(13) = 4) such that 44 = k + reverse(k).

Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067032, A067284.

A068064 a(n) = number of integers k such that palindrome A068062(n) = k + reverse(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 9, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 1, 10, 3, 4, 5, 6, 7, 8, 9, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 4, 8, 12, 16, 20, 24

Offset: 1

Klaus Brockhaus, Feb 16 2002

The number of representations of a palindrome as a + b, where b = reverse(a); if a = reverse(b) and a is different from b, then a + b and b + a count as different representations.

			a(9) = 4, since A068062(9) = 44 and for k = 13, 22, 31, 40 we have 44 = k + reverse(k).
a(16) = 9, since A068062(16) = 121 and for k = 29, 38, 47, 56, 65, 74, 83, 92, 110 we have 121 = k + reverse(k).

Cf. A002113, A067030, A067032, A068062.

Offset corrected by Sean A. Irvine, Jan 23 2024

A071914 Numbers n of the form k + reverse(k) for exactly three k.

Original entry on oeis.org

33, 176, 303, 323, 343, 363, 383, 403, 423, 443, 463, 483, 1221, 1616, 1636, 1656, 1676, 1696, 1716, 1736, 1756, 1776, 1796, 2761, 3003, 4983, 12021, 12421, 12621, 12821, 13021, 13221, 13421, 13621, 13821, 16016, 17996, 18238, 19778, 26161

Offset: 0

Klaus Brockhaus, Jun 13 2002

			33 = 12 + 21 = 21 + 12 = 30 + 03, 176 = 79 + 97 = 88 + 88 = 97 + 79, 383 = 142 + 241 = 241 + 142 = 340 + 043.

Index entries for sequences related to Reverse and Add!

Cf. A067030, A071265, A072040, A072041.

A072427 Numbers n for which there is a unique k such that n = k + reverse(k).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 101, 141, 161, 181, 198, 201, 221, 241, 261, 281, 1001, 1818, 1838, 1858, 1878, 1898, 1918, 1938, 1958, 1978, 1998, 2981, 10001, 10201, 10401, 10601, 10801, 11001, 11201, 11401, 11601, 11801, 18018, 19998

Offset: 1

Klaus Brockhaus, Jun 17 2002

Subsequence of A067030. First term is A072041(1). A068065 is a subsequence of this sequence.

			18 = 9 + 9; 261 = 180 + 081; 11401 = 10700 + 00701.

Cf. A067030, A068065, A072041, A072040, A071914.

var n,k,c,i,rev: integer; st,nst: string; end; m := 1; for n := 0 to 29000 do k := n div 2; c := 0; while k <= n and c < m + 1 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); if k mod 10 <> 0 and k <> rev then inc(c); end; end; inc(k); end; if c = m then write(n,","); end; end;

revAdd[n_] := n + FromDigits[Reverse[IntegerDigits[n]]]; ra=Table[revAdd[n], {n, 0, 10^5}]; t=Sort[Tally[ra]]; First /@ Select[t, #[[2]] == 1 && #[[1]] <= Length[ra] &]

A072428 Numbers n for which there are exactly four k such that n = k + reverse(k).

Original entry on oeis.org

44, 165, 404, 424, 444, 464, 484, 504, 524, 544, 564, 584, 1331, 1515, 1535, 1555, 1575, 1595, 1615, 1635, 1655, 1675, 1695, 2112, 2651, 3872, 4004, 5984, 13031, 13231, 13431, 13631, 13831, 14031, 14231, 14431, 14631, 14831, 15015, 16995

Offset: 1

Klaus Brockhaus, Jun 17 2002

Subsequence of A067030. First term is A072041(4).

Includes 4*(10^k+1) for k>=1. - Robert Israel, Jul 12 2019

			44 = k + reverse(k) for k = 13, 22, 31, 40; 1331 = k + reverse(k) for k = 1030, 1120, 1210, 1300.

Robert Israel, Table of n, a(n) for n = 1..441
Index entries for sequences related to Reverse and Add!

Cf. A067030, A072041.

var n,k,c,i,rev: integer; st,nst: string; end; m := 4; for n := 0 to 20000 do k := n div 2; c := 0; while k <= n and c < m + 1 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); if k mod 10 <> 0 and k <> rev then inc(c); end; end; inc(k); end; if c = m then write(n,","); end; end;

N:= 10^5:
revdigs:= proc(n) local L, i;
  L:= convert(n, base, 10);
  add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
V:= Vector(N):
for x from 1 to N do
  v:= x + revdigs(x);
  if v <= N then V[v]:= V[v]+1 fi;
od:
select(t -> V[t]=4, [$1..N]); # Robert Israel, Jul 12 2019

Offset changed by Robert Israel, Jul 12 2019

A072429 Numbers n for which there are exactly five k such that n = k + reverse(k).

Original entry on oeis.org

55, 154, 505, 525, 545, 565, 585, 605, 625, 645, 665, 685, 1414, 1434, 1441, 1454, 1474, 1494, 1514, 1534, 1554, 1574, 1594, 2541, 5005, 6985, 14014, 14041, 14241, 14441, 14641, 14841, 15041, 15241, 15441, 15641, 15841, 15994, 18458, 19558

Offset: 1

Klaus Brockhaus, Jun 17 2002

Subsequence of A067030. First term is A072041(5).

Contains 5*(10^k+1) for k>=1. - Robert Israel, Jul 12 2019

			55 = k + reverse(k) for k = 14, 23, 32, 41, 50; 1441 = k + reverse(k) for k = 1040, 1130, 1220, 1310, 1400.

Robert Israel, Table of n, a(n) for n = 1..254
Index entries for sequences related to Reverse and Add!

Cf. A067030, A072041.

var n,k,c,i,rev: integer; st,nst: string; end; m := 5; for n := 0 to 24600 do k := n div 2; c := 0; while k <= n and c < m + 1 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); if k mod 10 <> 0 and k <> rev then inc(c); end; end; inc(k); end; if c = m then write(n,","); end; end;

N:= 10^5:
revdigs:= proc(n) local L, i;
  L:= convert(n, base, 10);
  add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
V:= Vector(N):
for x from 1 to N do
  v:= x + revdigs(x);
  if v <= N then V[v]:= V[v]+1 fi;
od:
select(t -> V[t]=5, [$1..N]); # Robert Israel, Jul 12 2019

A072430 Numbers n for which there are exactly six k such that n = k + reverse(k).

Original entry on oeis.org

66, 143, 606, 626, 646, 666, 686, 706, 726, 746, 766, 786, 1313, 1333, 1353, 1373, 1393, 1413, 1433, 1453, 1473, 1493, 1551, 2222, 2431, 3113, 3762, 4873, 6006, 7986, 13013, 14993, 15051, 15251, 15451, 15651, 15851, 16051, 16126, 16251, 16451

Offset: 1

Klaus Brockhaus, Jun 17 2002

Subsequence of A067030. First term is A072041(6).

Includes 6*(10^k+1) for k >= 1. - Robert Israel, Jul 12 2019

			66 = k + reverse(k) for k = 15, 24, 33, 42, 51, 60; 626 = k + reverse(k) for k = 115, 214, 313, 412, 511, 610.

Robert Israel, Table of n, a(n) for n = 1..628
Index entries for sequences related to Reverse and Add!

Cf. A067030, A072041.

var n,k,c,i,rev: integer; st,nst: string; end; m := 6; for n := 0 to 17500 do k := n div 2; c := 0; while k <= n and c < m + 1 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); if k mod 10 <> 0 and k <> rev then inc(c); end; end; inc(k); end; if c = m then write(n,","); end; end;

N:= 10^5:
revdigs:= proc(n) local L, i;
  L:= convert(n, base, 10);
  add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
V:= Vector(N):
for x from 1 to N do
  v:= x + revdigs(x);
  if v <= N then V[v]:= V[v]+1 fi;
od:
select(t -> V[t]=6, [$1..N]); # Robert Israel, Jul 12 2019

Offset changed to 1 by Robert Israel, Jul 12 2019

cp's OEIS Frontend

A190481 Number of distinct integers with n digits which are the image of integers by the function Reverse and Add!.

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A227525 Squares which can be represented as sum of a prime and its reverse in at least one way.

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A335978 Numbers m of the form abs(k - reverse(k)) for at least one k.

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A067737 Integers n such that [number of integers k such that k is not of the form m + reverse(m) for any m and n occurs in the "Reverse and Add!" trajectory of k] is greater than [number of integers k such that n = k + reverse(k)].

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A068064 a(n) = number of integers k such that palindrome A068062(n) = k + reverse(k).

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A071914 Numbers n of the form k + reverse(k) for exactly three k.

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A072427 Numbers n for which there is a unique k such that n = k + reverse(k).

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A072428 Numbers n for which there are exactly four k such that n = k + reverse(k).

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ARIBAS

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A072429 Numbers n for which there are exactly five k such that n = k + reverse(k).

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