A067030 -id:A067030 - cp's OEIS frontend

A067031 Numbers that are not of the form k + reverse(k) for any k.

1, 3, 5, 7, 9, 13, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 0

Klaus Brockhaus, Dec 29 2001

			3 belongs to the sequence since there is no k such that k + reverse(k) = 3.

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000

Cf. A056964.
Complement of A067030.
Cf. A033865, A067030, A067032, A067033, A067034.

function a067031(a,b: integer); var k,n,i,rev: integer; test: boolean; st,nst: string; begin for n := a to b do k := 0; test := 1; while test and 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 test := 0; else inc(k); end; end; if k > n then write(n,","); end; end; end; a067031(0,100);

Module[{nn=100,rd},rd=Union[Table[n+FromDigits[ Reverse[ IntegerDigits[ n]]],{n,nn}]];Complement[Range[nn],rd]] (* Harvey P. Dale, Dec 16 2013 *)

A072041 a(n) is the smallest number of the form k + reverse(k) for exactly n integers k, or -1 if no such number exists.

Original entry on oeis.org

1, 0, 22, 33, 44, 55, 66, 77, 88, 99, 1111, -1, 2552, -1, 2662, 3443, 2772, -1, 2882, -1, 2992, 3663, -1, -1, 3773, 5445, -1, 3883, 4664, -1, 3993, -1, 4774, -1, -1, 5665, 4884, -1, -1, -1, 4994, -1, 6666, -1, -1, 5885, -1, -1, 6776, 7667, 5995, -1, -1, -1, 6886

Offset: 0

Klaus Brockhaus, Jun 08 2002

The negative terms are conjectural. Moreover they have a rather low degree of confirmation, since due to the time-consuming computations only numbers from 0 to 65000 have been taken into account. Random tests of larger numbers however seem to indicate, that only selected values of n occur. - In the cognate sequence A071266 two numbers a and b are counted only once, if n = a + b, a = reverse(b), b = reverse(a). Then 33 = 12 + 21 = 30 + 03 has a count of 2 and 44 = 13 + 31 = 22 + 22 = 40 + 04 has a count of 3, so 44 appears in A071266 instead of 33.

Terms are correct for k <= 10^8. - Sean A. Irvine, Aug 27 2024

			a(0) = 1, since 1 can in no way be written as k + reverse(k); a(1) = 0, since 0 = k + reverse(k) for k = 0; a(3) = 33, since 33 = k + reverse(k) for k = 12, 21 and 30.

Index entries for sequences related to Reverse and Add!

Cf. A067030, A071266, A072040.

A072040 Numbers n of the form k + reverse(k) for exactly two k.

Original entry on oeis.org

22, 187, 202, 222, 242, 262, 282, 302, 322, 342, 362, 382, 1717, 1737, 1757, 1777, 1797, 1817, 1837, 1857, 1877, 1897, 2002, 2871, 3982, 11211, 11411, 11611, 11811, 12011, 12211, 12411, 12611, 12811, 17017, 18128, 18997, 19888, 20002, 20202

Offset: 1

Klaus Brockhaus, Jun 08 2002

In the cognate sequence A071265 two numbers a and b are counted only once, if n = a + b, a = reverse(b), b = reverse(a). Therefore 187 = 89 + 98 = 98 + 89 does not appear in A071265.

			22 = 11 + 11 = 20 + 02, 187 = 89 + 98 = 98 + 89, 382 = 191 + 191 = 290 + 092.

Index entries for sequences related to Reverse and Add!

Cf. A067030, A071265, A072041, A096768.

# Maple program from N. J. A. Sloane, Mar 07 2016. Assumes digrev (from the "transforms" file) is available:
M:=21000; b := Array(1..M,0);
for n from 1 to M do
t1:=n+digrev(n);
if t1 <= M then b[t1]:=b[t1]+1; fi;
od:
ans:=[];
for n from 1 to M do
if b[n]=2 then ans:=[op(ans),n]; fi; od:
ans;

M = 10^5; digrev[n_] := IntegerDigits[n] // Reverse // FromDigits; Clear[b]; b[A072040%20=%20Reap%5BFor%5Bn%20=%201,%20n%20%3C=%20M,%20n++,%20If%5Bb%5Bn%5D%20==%202,%20Sow%5Bn%5D%5D%5D%5D%5B%5B2,%201%5D%5D%20(*%20_Jean-Fran%C3%A7ois%20Alcover">] = 0; For[n = 1, n <= M, n++, t1 = n + digrev[n]; If[t1 <= M, b[t1] = b[t1] + 1]]; A072040 = Reap[For[n = 1, n <= M, n++, If[b[n] == 2, Sow[n]]]][[2, 1]] (* _Jean-François Alcover, Oct 01 2016, after N. J. A. Sloane's Maple code *)

A067035 n sets a new record for the number of integers k such that n = k + reverse(k).

Original entry on oeis.org

0, 22, 33, 44, 55, 66, 77, 88, 99, 1111, 2552, 2662, 2772, 2882, 2992, 3663, 3773, 3883, 3993, 4774, 4884, 4994, 5885, 5995, 6886, 6996, 7887, 7997, 8888, 8998, 9889, 9999, 199991, 258852, 259952, 268862, 269962, 278872, 279972, 288882, 289982

Offset: 1

Klaus Brockhaus, Dec 29 2001

RECORDS transform of A067032. A067036 gives the corresponding records.

Are all terms palindromes? - David A. Corneth, Jan 16 2020

			33 belongs to the sequence because for three integers k (cf. A067032) we have 33 = k + reverse(k) and for m < 33 there are at most two integers j such that m = j + reverse(j).

Giovanni Resta, Table of n, a(n) for n = 1..125
N. J. A. Sloane, Transforms
Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067032, A067033, A067034, A067036.

Offset set to 1 by Giovanni Resta, Jan 16 2020

A067036 Records for the number of integers k such that an integer is of the form k + reverse(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 24, 27, 30, 32, 36, 40, 45, 50, 54, 60, 63, 70, 72, 80, 81, 90, 100, 108, 120, 126, 140, 144, 160, 162, 180, 200, 210, 216, 240, 243, 270, 300, 320, 324, 360, 400, 405, 450, 500, 540, 600, 630, 700, 720, 800, 810

Offset: 1

Klaus Brockhaus, Dec 29 2001

RECORDS transform of A067032. A067035 gives the corresponding integers at which these records are attained.

			3 is a record since there is an integer n (viz. 33, cf. A067032) such that for three integers k we have n = k + reverse(k) and for m < n there are at most two integers j such that m = j + reverse(j).

Giovanni Resta, Table of n, a(n) for n = 1..125
N. J. A. Sloane, Transforms
Index entries for sequences related to Reverse and Add!

Cf. A067030, A067031, A067032, A067033, A067034, A067035.

Offset set to 1 by Giovanni Resta, Jan 16 2020

A092213 In base 2: numbers n of the form k + reverse(k) for at least one k.

Original entry on oeis.org

0, 2, 3, 5, 6, 9, 10, 14, 15, 17, 18, 21, 24, 25, 27, 30, 33, 34, 35, 37, 42, 44, 45, 51, 52, 54, 57, 62, 63, 65, 66, 69, 75, 78, 81, 84, 85, 90, 93, 96, 99, 101, 102, 105, 108, 114, 115, 119, 121, 126, 129, 130, 133, 135, 139, 146, 149, 150, 153, 155, 164, 165, 166

Offset: 0

Klaus Brockhaus, Feb 25 2004

Base-2 analog of A067030 (base 10) and A091678 (base 4). Complement of A092214.

			25 is a term since 25 (decimal) = 11001 = 10100 + 00101 = 20 (decimal) + 5 (decimal).

Cf. A035522, A092214, A067030, A091678.

A358985 a(n) is the number of numbers of the form k + reverse(k) for at least one n-digit number k.

Original entry on oeis.org

10, 18, 180, 342, 3420, 6498, 64980, 123462, 1234620, 2345778, 23457780, 44569782, 445697820, 846825858, 8468258580, 16089691302, 160896913020, 305704134738, 3057041347380, 5808378560022, 58083785600220, 110359192640418, 1103591926404180, 2096824660167942

Offset: 1

Jon E. Schoenfield, Dec 08 2022

			There are 10 numbers of the form k + reverse(k) for 1-digit numbers k: 0, 2, 4, 6, 8, 10, 12, 14, 16, and 18, so a(1) = 10.
There are 18 numbers of the form k + reverse(k) for 2-digit numbers k: 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, and 198, so a(2) = 18.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,19).

Cf. A067030, A358986.

LinearRecurrence[{0, 19}, {10, 18, 180}, 30] (* Paolo Xausa, Oct 04 2024 *)

a(n) = (18+(n<2))*19^(n\2-1)*10^(n%2); \\ Jinyuan Wang, Dec 14 2022

def A358985(n):
    if n == 1:
        return 10
    kset = set()
    for i in range(10**(n-2),10**(n-1)):
        for j in range(int((s:=str(i))[0])+1):
            kset.add(10*i+j+int(str(j)+s[::-1]))
    return len(kset) # Chai Wah Wu, Dec 09 2022

a(2n+1) = 10*a(2n) for n > 0. - Chai Wah Wu, Dec 09 2022
a(2n) = 19*a(2n-2) for n > 1. - Robert G. Wilson v, Dec 10 2022
a(n) = 19*a(n-2). - Wesley Ivan Hurt, Mar 17 2023

More terms from Jinyuan Wang, Dec 14 2022

A367796 Primes p such that the sum of p and its reversal is the square of a prime.

Original entry on oeis.org

2, 29, 47, 83, 20147, 23117, 24107, 63113, 80141, 81131, 261104399, 262005299, 262104299, 262203299, 263302199, 264203099, 264302099, 264500099, 270401489, 271500389, 273104189, 273302189, 274401089, 282203279, 284302079, 284500079, 291104369, 291203369, 292005269, 293005169, 293104169, 294302069

Offset: 1

Robert Israel, Nov 30 2023

Terms > 83 have an odd number of digits and an even first digit.

			A056964(a(n)) = 121 = 11^2 for 2 <= n <= 4.
A056964(a(n)) = 94249 = 307^2 for 5 <= n <= 10.
A056964(a(n)) = 1254505561 = 35419^2 for 11 <= n <= 71.

David A. Corneth, Table of n, a(n) for n = 1..5743 (terms <= 10^15)
David A. Corneth, PARI program

Cf. A056964, A067030, A061783. Subset of A367793.

digrev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(t) local v;
  v:= sqrt(t+digrev(t));
  v::integer and isprime(v)
end proc:
R:= 2, 29, 47, 83: count:= 4: flag:= true:
for d from 3 to 9 by 2 do
  p:= prevprime(10^(d-1));
  for i from 1 do
    p:= nextprime(p);
    p1:= floor(p/10^(d-1));
    if p1::odd then p:= nextprime((p1+1)*10^(d-1)) fi;
    if p > 10^d then break fi;
    if filter(p) then
       count:= count+1; R:= R,p;
fi od od:
R;

Select[Prime[Range[10^6]], PrimeQ[Sqrt[#+FromDigits[Reverse[IntegerDigits[#]]]]] &] (* Stefano Spezia, Dec 10 2023 *)

PARI
```
\\ See PARI link
```

A068062 Palindromes n of the form k + reverse(k) for at least one k.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 121, 141, 161, 181, 202, 222, 242, 262, 282, 303, 323, 343, 363, 383, 404, 424, 444, 464, 484, 505, 525, 545, 565, 585, 606, 626, 646, 666, 686, 707, 727, 747, 767, 787, 808, 828, 848, 868, 888, 909, 929

Offset: 1

Klaus Brockhaus, Feb 15 2002

Intersection of A002113 and A067030.

			33 belongs to this sequence, since 12 + 21 = 33 (cf. A067030).

Cf. A002113, A067030, A068061.

A305130 Numbers k with the property that there exists a positive integer M, called multiplier, such that the sum of the digits of k times the multiplier added to the reversal of this product gives k.

Original entry on oeis.org

10, 11, 12, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 121, 132, 141, 165, 181, 201, 202, 221, 222, 261, 262, 282, 302, 303, 322, 323, 342, 343, 363, 403, 404, 423, 424, 444, 463, 483, 504, 505, 525, 545, 564, 584, 585, 605, 606, 645, 646, 666, 686, 706

Offset: 1

Viorel Nitica, May 26 2018

These numbers are related to the taxicab number 1729. This is why they might be called "additive Hardy-Ramanujan numbers".

			For k = 11 the sum of the digits is 2 and the multiplier is 5: 2 * 5 = 10 and 10 + 01 = 11.
For k = 747 the sum of the digits is 18 and the multiplier is 7: 18 * 7 = 126 and 126 + 621 = 747.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Viorel Nitica, About some relatives of the taxicab numbers, submitted to Journal of Integer Sequences, (2018). [Where these numbers are introduced.]
Viorel Niţică, Jeroz Makhania, About the Orbit Structure of Sequences of Maps of Integers, Symmetry (2019), Vol. 11, No. 11, 1374.

Subsequence of A067030.

Cf. A004086, A011541, A305131.

Block[{k, d, j}, Reap[Do[k = 1; d = Total@ IntegerDigits[i]; While[Nor[k > i, Set[j, # + IntegerReverse@ #] == i &[d k]], k++]; If[j == i, Sow[{i, k}]], {i, 720}]][[-1, 1, All, 1]] ] (* Michael De Vlieger, Jan 28 2020 *)

cp's OEIS Frontend

A067031 Numbers that are not of the form k + reverse(k) for any k.

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A072041 a(n) is the smallest number of the form k + reverse(k) for exactly n integers k, or -1 if no such number exists.

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A072040 Numbers n of the form k + reverse(k) for exactly two k.

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A067035 n sets a new record for the number of integers k such that n = k + reverse(k).

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A067036 Records for the number of integers k such that an integer is of the form k + reverse(k).

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A092213 In base 2: numbers n of the form k + reverse(k) for at least one k.

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A358985 a(n) is the number of numbers of the form k + reverse(k) for at least one n-digit number k.

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A367796 Primes p such that the sum of p and its reversal is the square of a prime.

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