cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A356648 Numbers whose square is of the form k + reversal of digits of k, for some k.

Original entry on oeis.org

2, 4, 11, 22, 25, 33, 101, 121, 141, 202, 222, 264, 303, 307, 451, 836, 1001, 1111, 1221, 1232, 2002, 2068, 2112, 2222, 2305, 2515, 2626, 2636, 2776, 3003, 3958, 3972, 4015, 4081, 7975, 8184, 9757, 10001, 10201, 10401, 11011, 11121, 11211, 12012, 12021, 12221, 13046, 16581, 20002
Offset: 1

Views

Author

Nicolay Avilov, data a(10)-a(37) from Oleg Sorokin, Dec 10 2022

Keywords

Comments

Square roots of the squares in A067030.

Examples

			4 is a term since 4^2 = 16 = 8 + 8;
11 is a term since 11^2 = 121 = 29 + 92 is sum of k=29 and its reversal 92;
22 is a term since 22^2 = 484 = 143 + 341;
10201 is a term since 10201^2 = 104060401 = 100030400 + 4030001.

Links

Crossrefs

Cf. A056964, A061230, A067030, A358880, A358984.

Programs

  • PARI
    L=vectorsmall(100000);
\\ Takes a few minutes of CPU time
for (k=1, 2*10^8, my(d=digits(k), r=fromdigits(Vecrev(d)), s); if (issquare(k+r, &s), L[s]=1));
for (k=1, 21000, if(L[k], print1(k,", "))) \\ Hugo Pfoertner, Dec 13 2022
(C++, Haskell) See Code Golf link.

Formula

a(n) = sqrt(A358880(n)). - Michel Marcus, Dec 25 2022

Extensions

a(38) and beyond from Hugo Pfoertner, Dec 12 2022

A358986 a(n) is the number of numbers of the form k + reverse(k) for at least one number k < 10^n.

Original entry on oeis.org

10, 28, 207, 548, 3966, 10462, 75435, 198890, 1433489, 3779246
Offset: 1

Views

Author

Jon E. Schoenfield, Dec 08 2022

Keywords

Comments

This sequence differs from the partial sums of A358985; see the Example section.

Examples

			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 -- and none of these 18 numbers are among the 10 numbers counted in a(1), so a(2) = 10 + 18 = 28.
There are 180 numbers of the form k + reverse(k) for 3-digit numbers k, but exactly one of those -- 121 = 110 + reverse(110) = 110 + 11 -- is also a number of the form k + reverse(k) for a 2-digit number k: e.g., 29 + reverse(29) = 29 + 92 = 121. So a(3) = 10 + 18 + 180 - 1 = 207.

Crossrefs

Cf. A067030, A358985.

Programs

  • Python
    def A358986(n):
    kset = set()
    for i in range(1,10**(n-1)):
        for j in range(int((s:=str(i))[0])+1):
            kset.add(10*i+j+int(str(j)+s[::-1]))
    return 10+len(kset) # Chai Wah Wu, Dec 09 2022

Extensions

a(8)-a(10) from Chai Wah Wu, Dec 09 2022

A067287 n sets a new record for the 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 (cf. A067284).

Original entry on oeis.org

0, 22, 33, 44, 66, 88, 110, 121, 242, 484, 968, 1837, 2222, 3102, 4444, 4884, 7106, 8888, 12221, 24442, 44044, 48884, 88088, 176176, 293392, 295482, 466664, 597795, 688886, 711106, 797797, 930028, 933328, 997799, 1112111, 1197801, 1686861
Offset: 0

Views

Author

Klaus Brockhaus, Feb 04 2002

Keywords

Comments

A067288 gives the corresponding records.

Examples

			33 belongs to the sequence because three integers k (viz. 3, 21, 30) are not of the form j + reverse(j) for any j and 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.

Links

Crossrefs

Cf. A067030, A067031, A067032, A067035, A067036, A067284, A067288.

Extensions

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

A068065 Palindromes n for which there is a unique k such that n = k + reverse(k).

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 101, 141, 161, 181, 1001, 10001, 10201, 10401, 10601, 10801, 100001, 1000001, 1002001, 1004001, 1006001, 1008001, 10000001, 100000001, 100020001, 100040001, 100060001, 100080001, 1000000001, 10000000001
Offset: 1

Views

Author

Klaus Brockhaus, Feb 16 2002

Keywords

Comments

Subsequence of A068062; A068062(k) is in this sequence if and only if A068064(k) = 1. At first sight, 121 seems to be missing, but in fact 121 does not belong here (cf. example in A068064).

Examples

			10601 is in the sequence, since 10601 = 10300 + 00301 and for no other k we have 10601 = k + reverse(k).

Crossrefs

Cf. A002113, A067030, A067032, A068062, A068064.

A071265 Numbers which can be written in exactly two different ways as k + R(k) where R(k) is k reversed (A004086).

Original entry on oeis.org

22, 33, 165, 176, 202, 222, 242, 262, 282, 302, 303, 322, 323, 342, 343, 362, 363, 382, 383, 403, 423, 443, 463, 483, 1515, 1535, 1555, 1575, 1595, 1615, 1616, 1635, 1636, 1655, 1656, 1675, 1676, 1695, 1696, 1716, 1736, 1756, 1776, 1796, 2002, 2871, 3003
Offset: 1

Views

Author

Amarnath Murthy, Jun 01 2002

Keywords

Comments

The sums are unordered, so for example 12 + 21 is not counted as distinct from 21 + 12. - Sean A. Irvine, Jul 06 2024

Examples

			22 = 11 + 11 = 20 + 02, 202 =101 + 101 = 200 + 002.

Links

Crossrefs

Cf. A071264, A071266, A067030, A067032, A067033, A067034.

Extensions

More terms from Vladeta Jovovic and Klaus Brockhaus, Jun 03 2002
Offset corrected by Sean A. Irvine, Jul 06 2024

A088169 Numbers n such that the set S(n) = {k: k + reverse(k) = n} is not empty and all elements of S(n) have the same number of digits as n.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 141, 161, 181, 201, 202, 221, 222, 241, 242, 261, 262, 281, 282, 302, 303, 322, 323, 342, 343, 362, 363, 382, 383, 403, 404, 423, 424, 443, 444, 463, 464, 483, 484, 504, 505, 524, 525, 544, 545, 564, 565
Offset: 1

Views

Author

Klaus Brockhaus, Sep 21 2003

Keywords

Comments

Subsequence of A067030.

Examples

			404 is a term since S(404) = {103, 202, 301, 400}.

Links

Crossrefs

Cf. A067030, A088170, A088171.

A088170 Numbers n such that the set S(n) = {k: k + reverse(k) = n} is not empty and all elements of S(n) have one digit less than n has.

Original entry on oeis.org

10, 12, 14, 16, 18, 110, 132, 143, 154, 165, 176, 187, 198, 1009, 1010, 1029, 1030, 1049, 1050, 1069, 1070, 1089, 1090, 1110, 1130, 1131, 1150, 1151, 1170, 1171, 1190, 1191, 1211, 1212, 1231, 1232, 1251, 1252, 1271, 1272, 1291, 1292, 1312, 1313, 1332
Offset: 1

Views

Author

Klaus Brockhaus, Sep 21 2003

Keywords

Comments

Subsequence of A067030.
The most significant digit of all terms is 1.

Examples

			165 is a term since S(165) = {69, 78, 87, 96}.

Links

Crossrefs

Cf. A067030, A088169, A088171.

A088171 Numbers n such that the set S(n) = {k: k + reverse(k) = n} is not empty, at least one element of S(n) has the same number of digits as n and at least one element of S(n) has one digit less than n has.

Original entry on oeis.org

121, 1111, 11011, 12221, 110011, 121121, 1100011, 1112111, 1197801, 1208911, 1210121, 1211111, 1222221, 11000011, 11111111, 11988801, 12098911, 12100121, 12101111, 12211221, 110000011, 110121011, 111089011, 111101111, 111111011
Offset: 1

Views

Author

Klaus Brockhaus, Sep 21 2003

Keywords

Comments

Subsequence of A067030.
The most significant digit of all terms is 1.

Examples

			1111 is a term since S(1111) = {209, 308, 407, 506, 605, 704, 803, 902, 1010, 1100}.

Links

Crossrefs

Cf. A067030, A088169, A088170.

A091678 In base 4, numbers n of the form k + reverse(k) for at least one k.

Original entry on oeis.org

0, 2, 4, 5, 6, 10, 15, 17, 20, 25, 30, 33, 34, 41, 42, 50, 51, 58, 59, 65, 67, 68, 75, 76, 84, 85, 92, 93, 101, 102, 105, 109, 110, 118, 125, 126, 130, 145, 150, 165, 170, 185, 190, 195, 210, 215, 230, 235, 250, 255, 257, 260, 275, 280, 289, 295, 300, 315, 320, 321
Offset: 0

Views

Author

Klaus Brockhaus, Jan 28 2004

Keywords

Comments

Base-4 analog of A067030. Complement of A091679.

Examples

			25 is a term since 25 (decimal) = 121 = 110 + 011 = 20 (decimal) + 5 (decimal).

Crossrefs

Cf. A035524, A091679, A067030.

A096768 Numbers n of the form k + reverse(k) for two or more values of k.

Original entry on oeis.org

22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 202, 222, 242, 262, 282, 302, 303, 322, 323, 342, 343, 362, 363, 382, 383, 403, 404, 423, 424, 443, 444, 463, 464, 483, 484, 504, 505, 524, 525, 544, 545, 564, 565, 584, 585, 605, 606
Offset: 1

Views

Author

Chuck Seggelin (seqfan(AT)plastereddragon.com), Jul 08 2004

Keywords

Examples

			22 belongs to the sequence since 11 + 11 = 22 and 20 + 2 = 22 (k = {11, 20}); 33 belongs to the sequence since 12 + 21 = 33, 21 + 12 = 33 and 30 + 3 = 33 (k = {12, 21, 30}).

Links

Crossrefs

Cf. A067030, A072040 (exactly two values of k).

Programs

  • Maple
    reverse:= proc (d) local n,m; m:=0;n:=d; while n>0 do m:=m*10+(n mod 10); n:=(n-(n mod 10))/10; od; m; end; P:={};P2:={};for i to 5000 do; if i>0 then; r:=i+reverse(i); rat:={r}; if P intersect rat = {} then P:=P union rat else P2:=P2 union rat fi; fi; od; P2;
# Maple program from N. J. A. Sloane, Mar 07 2016. Assumes digrev (from the "transforms" file) is available:
M:=1000; 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;
  • Mathematica
    M = 10^3; digrev[n_] := IntegerDigits[n] // Reverse // FromDigits; Clear[b]; b[A096768%20=%20Reap%5BFor%5Bn%20=%201,%20n%20%3C=%20M,%20n++,%20If%5Bb%5Bn%5D%20%3E=%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]]; A096768 = 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 *)
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