A004086 -id:A004086 - cp's OEIS frontend

A278328 Numbers n such that abs(n - rev(n)) is a square, where rev(n) is the decimal reverse of n (A004086).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 21, 22, 23, 26, 32, 33, 34, 37, 40, 43, 44, 45, 48, 51, 54, 55, 56, 59, 62, 65, 66, 67, 73, 76, 77, 78, 84, 87, 88, 89, 90, 95, 98, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262

Offset: 1

Jonathan Frech, Nov 18 2016

All palindromes are in this sequence, hence it is infinite.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

A002113 is a subsequence.

Cf. A000290, A004086, A016766, A056965.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, -1, a(n-1)) while not issqr(abs(k-(s->
       parse(cat(s[-i]$i=1..length(s))))(""||k))) do od: k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 18 2016

Select[Range@ 262, IntegerQ@ Sqrt@ Abs[# - FromDigits@ Reverse@ IntegerDigits@ #] &] (* Michael De Vlieger, Nov 18 2016 *)

is(n) = issquare(abs(n - eval(concat(Vecrev(Str(n)))))) \\ Felix Fröhlich, Nov 18 2016

is(n, {b=10}) = issquare(abs(n - subst(Polrev(digits(n, b),'x),'x,b))); \\ Gheorghe Coserea, Nov 27 2016

import math
n = 0
while True:
    if math.sqrt(abs(n-int(str(n)[::-1])))%1 == 0:
        print(n)
    n += 1 # Jonathan Frech, Nov 18 2016

A062390 Numbers k such that (k + R(k)) / (k - R(k)) = +-11 where R(k) is the digit reversal of k (A004086).

Original entry on oeis.org

45, 54, 495, 594, 4545, 4995, 5454, 5994, 45045, 49995, 54054, 59994, 450045, 454545, 495495, 499995, 540054, 545454, 594594, 599994, 4500045, 4549545, 4950495, 4999995, 5400054, 5459454, 5940594, 5999994, 45000045, 45045045

Offset: 1

Amarnath Murthy, Jun 27 2001

Are there numbers for which (k + R(k)) / (k - R(k)) is a number other than 11?

			(5994 + 4995) /(5994 - 4995) = 10989/999 = 11, so 5994 is in the sequence.

Harry J. Smith, Table of n, a(n) for n = 1..44

dr11Q[n_]:=Module[{dr=FromDigits[Reverse[IntegerDigits[n]]]},n!=dr && Abs[(n+dr)/(n-dr)]==11]; Select[Range[45100000],dr11Q] (* Harvey P. Dale, Oct 03 2011 *)

{ n=0; for (m=1, 10^9, x=m; r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); if ((m + r) == 11*abs(m - r), write("b062390.txt", n++, " ", m); if (n==44, break)) ) } \\ Harry J. Smith, Aug 07 2009

Corrected formula and more terms from Jason Earls, Jun 29 2001
Definition corrected and incorrect formula deleted by Harry J. Smith, Aug 06 2009
Missing terms adding like a(5) = 4545 by Harry J. Smith, Aug 07 2009

A068159 a(n) = floor[ n/R(n) ], where R(n) (A004086) = Digit reversal of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 1, 0, 0, 0, 0, 0, 0, 0, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 10, 2, 1, 1, 1, 0, 0, 0, 0, 0, 10, 3, 2, 1, 1, 1, 0, 0, 0, 0, 10, 3, 2, 1, 1, 1, 1, 0, 0, 0, 10, 4, 2, 1, 1, 1, 1, 1, 0, 0, 10, 4, 2, 2, 1, 1, 1, 1, 1, 0, 10, 4, 3, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amarnath Murthy, Feb 24 2002

			a(61) = floor[61 / 16] = 3.

Reversal = ToExpression @ StringReverse @ ToString[ # ] &; Table[ Floor[n/Reversal[n]], {n, 1, 100}]
Table[Floor[n/IntegerReverse[n]],{n,100}] (* Harvey P. Dale, Mar 24 2025 *)

Edited by Robert G. Wilson v, Mar 02 2002

A072017 Numbers k such that gcd(k, reverse(k)) = 81 = 3^4, where reverse(x) = A004086(x).

Original entry on oeis.org

2899999989, 2989999899, 2999889999, 3799999899, 3898989999, 3899799999, 3899999988, 3979989999, 3988899999, 3989999898, 3989999979, 3998999889, 3999889998, 3999898989, 3999899799, 3999979989, 3999988899, 4699998999

Offset: 1

Labos Elemer, Jun 05 2002

Numerous solutions can be constructed by inserting strings of suitable digits between digits of terms in A071016.

			k = 3*3*3*3*3*449*64157 and reverse(k) = 2*2*3*3*3*3*31*67*14827, GCD = 81.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A004086, A055483, A069554, A071686, A072005, A072016, A072018.

A074163 Smallest k, not divisible by 10, such that R(k) > n*k, where R(k) is the digit reversal of k (A004086).

Original entry on oeis.org

12, 13, 15, 17, 19, 107, 108, 109

Offset: 1

Amarnath Murthy, Aug 30 2002

			a(3) = 15, 51 > 3*15, a(3) is not 14 as 41 < 42 = 3*14.

Cf. A074164.

A074813 Number of primes between n and R(n) (inclusive), where R(n) (A004086) is the digit reversal of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 4, 1, 3, 6, 7, 9, 12, 14, 15, 17, 8, 3, 0, 3, 4, 6, 9, 11, 13, 15, 9, 6, 3, 0, 3, 5, 7, 10, 11, 12, 10, 7, 4, 3, 0, 2, 4, 7, 8, 9, 13, 9, 6, 5, 2, 0, 2, 5, 7, 8, 14, 12, 9, 7, 4, 2, 0, 3, 4, 5, 16, 14, 11, 10, 7, 5, 3, 0, 2, 4, 18, 15, 13, 11, 8, 7, 4, 2, 0, 2, 20, 17

Offset: 1

Jason Earls, Sep 08 2002

A060568 is the exclusive version of this sequence.

			The primes between n = 13 and 31, inclusive, are 13, 17, 19, 23, 29, 31; so a(13) = 6.

A074858 a(n) = a(n-1) + a(n-2) + R(a(n-3)) where a(0) = a(1) = a(2) = 1 and R(n) (A004086) means the reverse of n.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 31, 57, 159, 229, 463, 1643, 3028, 5035, 11524, 24762, 41591, 108864, 177197, 305575, 951573, 2048919, 3575995, 6000073, 18774470, 30770296, 53244772, 91462849, 213915324, 333122408, 641864151, 1398505871, 2844591355

Offset: 0

Felice Russo, Sep 11 2002

			a(9) = 57 + 31 + R(17) = 57 + 31 + 71 = 159.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000213.

R:=proc(n) local nn, nnn: nn:=convert(n,base,10): add(nn[nops(nn)+1-j]*10^(j-1),j=1..nops(nn)) end: a[0]:=1: a[1]:=1: a[2]:=1: for n from 3 to 34 do a[n]:=a[n-1]+a[n-2]+R(a[n-3]) od: seq(a[n],n=0..34); # Emeric Deutsch, Jul 25 2005

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==a[n-1]+a[n-2]+IntegerReverse[ a[n-3]]},a,{n,40}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 05 2020 *)

More terms from Emeric Deutsch, Jul 25 2005

A074860 a(n) = a(n-1) + R(a(n-2)) + R(a(n-3)) where a(0)=a(1)=a(2)=1 and R(k) = A004086(k) is the reverse of k.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 31, 111, 195, 319, 1021, 2525, 4639, 11092, 25708, 64083, 173846, 292644, 979061, 2073724, 2680995, 7115676, 17380240, 30136219, 41109707, 136581181, 298634398, 550610143, 1625232666, 2859685613, 9863026929, 19691221772, 32153295043

Offset: 0

Felice Russo, Sep 11 2002

Cf. A000213, A004086.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
a:= proc(n) option remember; `if`(n<3, 1,
      a(n-1) + R(a(n-2)) + R(a(n-3)))
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Jun 17 2021

Corrected and extended by David Garber, Oct 23 2002

Offset corrected by and more terms from Alois P. Heinz, Jun 17 2021

A074862 Iccanartet sequence: a(n)=R[a(n-1)]+R[a(n-2)]+R[a(n-3)]+R[a(n-4)] where a(1)=a(2)=a(3)=a(4)=1 and R(n) (A004086) is the reverse of n.

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 13, 43, 76, 139, 1063, 4633, 7963, 11593, 50173, 83677, 157951, 314005, 774907, 1447279, 11097082, 39016342, 62877022, 84245371, 91872178, 150920986, 815588944, 1243396636

Offset: 1

Felice Russo, Sep 11 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000288.

Transpose[NestList[Join[Rest[#],{Total[FromDigits/@(Reverse/@ (IntegerDigits/@#))]}]&,{1,1,1,1},30]][[1]] (* Harvey P. Dale, May 02 2012 *)

More terms from David Garber, Oct 23 2002

A074863 a(n) = a(n-1) + a(n-2) + a(n-3) + R(a(n-4)) where a(0)=a(1)=a(2)=a(3)=1 and R(n) (A004086) is the reverse of n.

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 13, 25, 49, 94, 199, 394, 781, 1423, 3589, 6286, 11485, 24601, 52225, 95137, 230374, 388378, 766114, 1458025, 3085549, 6183571, 11138812, 25616473, 52394659, 90903760, 190798003, 371558074, 748909162, 1317996148, 2739360475, 5277120958

Offset: 0

Felice Russo, Sep 11 2002

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A000288.

read transforms; A074863 := proc(n) option remember; local i,j,k,t1; if n <= 3 then 1 else A074863(n-1)+A074863(n-2)+A074863(n-3) +digrev(A074863(n-4)); fi; end;

Corrected and extended by David Garber, Oct 23 2002

cp's OEIS Frontend

A278328 Numbers n such that abs(n - rev(n)) is a square, where rev(n) is the decimal reverse of n (A004086).

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A062390 Numbers k such that (k + R(k)) / (k - R(k)) = +-11 where R(k) is the digit reversal of k (A004086).

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A068159 a(n) = floor[ n/R(n) ], where R(n) (A004086) = Digit reversal of n.

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A072017 Numbers k such that gcd(k, reverse(k)) = 81 = 3^4, where reverse(x) = A004086(x).

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A074163 Smallest k, not divisible by 10, such that R(k) > n*k, where R(k) is the digit reversal of k (A004086).

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A074813 Number of primes between n and R(n) (inclusive), where R(n) (A004086) is the digit reversal of n.

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A074858 a(n) = a(n-1) + a(n-2) + R(a(n-3)) where a(0) = a(1) = a(2) = 1 and R(n) (A004086) means the reverse of n.

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A074860 a(n) = a(n-1) + R(a(n-2)) + R(a(n-3)) where a(0)=a(1)=a(2)=1 and R(k) = A004086(k) is the reverse of k.

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A074862 Iccanartet sequence: a(n)=R[a(n-1)]+R[a(n-2)]+R[a(n-3)]+R[a(n-4)] where a(1)=a(2)=a(3)=a(4)=1 and R(n) (A004086) is the reverse of n.

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