A004086 -id:A004086 - cp's OEIS frontend

A070838 Smallest prime p such that |p - R(p)| = 9n, where R(n) is digit reversal of n, A004086; or 0 if no such prime exists.

23, 13, 41, 37, 61, 17, 29, 19, 0, 1231, 211, 0, 0, 0, 0, 0, 0, 0, 0, 1021, 2903, 103, 0, 0, 0, 0, 0, 0, 0, 1031, 2081, 0, 401, 0, 0, 0, 0, 0, 0, 1151, 4073, 0, 0, 307, 0, 0, 0, 0, 0, 1051, 2281, 0, 0, 0, 227, 0, 0, 0, 0, 1061, 2161, 0, 0, 0, 0, 107, 0, 0, 0, 1181, 2371, 0, 0, 0, 0, 0

Offset: 1

Amarnath Murthy, May 12 2002

Cf. A070837.

Corrected and extended by Sascha Kurz, Jan 02 2003

A071240 Arithmetic mean of k and R(k) where k is a number using only odd digits and R(k) is its digit reversal (A004086).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 22, 33, 44, 55, 22, 33, 44, 55, 66, 33, 44, 55, 66, 77, 44, 55, 66, 77, 88, 55, 66, 77, 88, 99, 111, 212, 313, 414, 515, 131, 232, 333, 434, 535, 151, 252, 353, 454, 555, 171, 272, 373, 474, 575, 191, 292, 393, 494, 595, 212, 313, 414, 515, 616

Offset: 0

Amarnath Murthy, May 20 2002

Cf. A004086, A014261, A071241, A071242.

reversal := proc(n) local i, len, new, temp:
    new := 0: temp := n: len := floor(log[10](n+.1))+1:
    for i from 1 to len do
        new := new+irem(temp, 10)*10^(len-i):
        temp := floor(temp/10):
    od:
    RETURN(new):
end:
allodd := proc(n) local i, flag, len, temp:
    temp := n: flag := 1: if n=0 then flag := 0 fi: len := floor(log[10](n+.1))+1:
    for i from 1 to len do
        if irem(temp, 10) mod 2 = 1 then temp := floor(temp/10) else flag := 0 fi:
    od:
    RETURN(flag):
end:
for n from 1 to 501 by 2 do
    if allodd(n) = 1 then printf(`%d, `, (n+reversal(n))/2) fi:
od: # James Sellers, May 28 2002

{k + R(k)}/2 where k uses only odd digits 1, 3, 5, 7 and 9.

a(n) = (A014261(n) + A004086(A014261(n))) / 2. - Sean A. Irvine, Jul 06 2024

More terms from James Sellers, May 28 2002

A071242 Arithmetic mean of n and R(n) where n is a number such that the least significant digit and the most significant digits are of same parity and R(n) is its digit reversal (A004086).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 11, 22, 33, 44, 55, 22, 33, 44, 55, 66, 22, 33, 44, 55, 66, 33, 44, 55, 66, 77, 33, 44, 55, 66, 77, 44, 55, 66, 77, 88, 44, 55, 66, 77, 88, 55, 66, 77, 88, 99, 101, 202, 303, 404, 505, 111, 212, 313, 414, 515, 121

Offset: 0

Amarnath Murthy, May 20 2002

Cf. A071240, A071241.

reversal := proc(n) local i, len, new, temp: new := 0: temp := n: len := floor(log[10](n+.1))+1: for i from 1 to len do new := new+irem(temp, 10)*10^(len-i): temp := floor(temp/10): od: RETURN(new): end: for n from 0 to 500 do if type((n+reversal(n))/2, integer) then printf(`%d,`,(n+reversal(n))/2) fi: od: # James Sellers, May 28 2002

{n + R(n)}/2 for all n which yield integers.

More terms from James Sellers, May 28 2002

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

Amarnath Murthy, Jun 01 2002

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

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

Index entries for sequences related to Reverse and Add!

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

More terms from Vladeta Jovovic and Klaus Brockhaus, Jun 03 2002

Offset corrected by Sean A. Irvine, Jul 06 2024

A071266 Smallest numbers which can be written in exactly n different ways as k + R(k) where R(k) is k reversed (A004086).

Original entry on oeis.org

1, 0, 22, 44, 66, 88, 1111, 1661, 1771, 1881, 1991, 2662, 2772, 4444, 2882, 2992, 3773

Offset: 0

Amarnath Murthy, Jun 01 2002

a(17), if it exists, exceeds 10^10. - Sean A. Irvine, Jul 06 2024

			a(0) = 1, since 1 cannot be written as k + R(k); a(1) = 0, since 0 = k + R(k) for k = 0.
a(4) = 66 and the four partitions are (60,06),(51,15),(42,24),(33,33).

Index entries for sequences related to Reverse and Add!

More terms from Klaus Brockhaus, Jun 05 2002

A071687 Non-palindromic numbers such that either x=q1.Rev[x] or Rev[x]=q2.x, where R[x]=A004086[x] and q1 or q2 are integers not divisible by 10.

Original entry on oeis.org

510, 540, 810, 1089, 2100, 2178, 4200, 5200, 5610, 5700, 5940, 6300, 8400, 8712, 8910, 9801, 10989, 21978, 23100, 27000, 46200, 51510, 52200, 52800, 54540, 56610, 57200, 59940, 65340, 69300, 81810, 87912, 89910, 98901, 109989, 212100, 217800

Offset: 1

Labos Elemer, Jun 03 2002

			Includes special cases of A071685. Examples represented by {n, Rev[n], integer-quotient} triples: {1089, 9801, 9}, {87912, 21979, 4}, {5610, 165, 34}, {610000, 16, 38125}, etc.

Cf. A071685, A004046, A055483, A069554.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[ed[x]] Do[s=Mod[ma=Max[{n, tn[red[n]]}], mi=Min[{n, r=tn[red[n]]}]]; If[Equal[s, 0]&&!Equal[n, r] &&!Equal[Mod[ma/mi, 10], 0], Print[{n, r, Max[r/n, n/r]}]], {n, 1, 1000000}]

q=Max[n/Rev[n], Rev[n]/n]=10m+r integer, where r>0, q>1.

A072032 a(n) = gcd(2^n, reverse(2^n)) = gcd(2^n, A004086(2^n)) = A055483(2^n).

Original entry on oeis.org

2, 4, 8, 1, 1, 2, 1, 4, 1, 1, 2, 8, 2, 1, 1, 4, 1, 2, 1, 1, 2, 2, 2, 1, 1, 16, 1, 2, 1, 1, 4, 4, 2, 1, 1, 2, 1, 8, 1, 1, 8, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 8, 4, 1, 1, 1, 1, 1, 8, 1, 1, 16, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 8, 1, 1, 1, 1, 1, 1, 8, 1, 2, 2, 1, 1, 1, 1, 1, 1, 16, 1, 8, 1, 1

Offset: 1

Labos Elemer, Jun 07 2002

			n=12: a(12) = gcd(4096,6904) = 8.

Indranil Ghosh, Table of n, a(n) for n = 1..31180

Cf. A004086, A055483, A071686.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] red[x_] := Reverse[IntegerDigits[x]] Table[GCD[2^w, tn[red[2^w]]], {w, 1, 128}]

A074164 Smallest k such that R(k) > n*k, where R(k) is the digit reversal of k (A004086) (the reversal of 10 is taken to be 01 = 1, etc.).

Original entry on oeis.org

12, 13, 15, 17, 106, 107, 108, 109, 1099

Offset: 1

Amarnath Murthy, Aug 30 2002

As R(k) doesn't increase the number of digits, R(k)<10k and so the sequence is complete. - Sascha Kurz, Jan 15 2003

			a(3) = 15, 51 > 3*15, a(3) is not 14 as 41 < 42 = 3*14. a(12) = 430 > 12*34.
a(4) = 17 as 71 > 17*4 but 61 is < 16*4.

Cf. A074163.

P := proc(Nlo, Nhi,Klo,Khi) local A::list,k,n,d,s; d := (X::posint)->convert(X,base,10):s := (L::list)->sum(L[i]*10^(nops(L)-i),i=1..nops(L)):k := Klo:A := [seq(0,i=1..Nhi-Nlo+1)]: for n from Nlo to Nhi do while k

Edited by Sascha Kurz, Jan 15 2003

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 21 2007

A074861 Iccanobirt sequence: a(n) = R(a(n-1)) + R(a(n-2)) + R(a(n-3)) where a(1)=a(2)=a(3)=1 and R(n) (A004086) is the reverse of n.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 85, 138, 960, 958, 1759, 10499, 109831, 247873, 617044, 958359, 1773317, 8528346, 14525888, 102424570, 170715000, 164793813, 394338733, 656748025, 1177078610, 1027388860, 1378392568, 9510483643, 12805616091

Offset: 1

Felice Russo, Sep 11 2002

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

Cf. A000213.

nxt[{a_,b_,c_}]:={b,c,FromDigits[Reverse[IntegerDigits[a]]] + FromDigits[ Reverse[IntegerDigits[b]]] + FromDigits[ Reverse[ IntegerDigits[c]]]}; Transpose[NestList[nxt,{1,1,1},30]][[1]] (* Harvey P. Dale, Nov 13 2012 *)

More terms from David Garber, Oct 23 2002

Definition adapted to offset by Georg Fischer, Jun 18 2021

A077337 Numbers k such that k and R(k) both are squarefree where R(n) (A004086) is the digit reversal of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 26, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 47, 51, 53, 55, 58, 59, 62, 66, 70, 71, 73, 74, 77, 78, 79, 83, 85, 87, 91, 93, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 127, 129

Offset: 1

Amarnath Murthy, Nov 04 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 6.

Cf. A004086, A005117.

Select[Range[200],And@@SquareFreeQ/@{#,FromDigits[Reverse[ IntegerDigits[ #]]]}&] (* Harvey P. Dale, Jan 04 2014 *)
Select[Range[200],AllTrue[{#,IntegerReverse[#]},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2018 *)

More terms from David Wasserman, Oct 25 2006

cp's OEIS Frontend

A070838 Smallest prime p such that |p - R(p)| = 9n, where R(n) is digit reversal of n, A004086; or 0 if no such prime exists.

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A071240 Arithmetic mean of k and R(k) where k is a number using only odd digits and R(k) is its digit reversal (A004086).

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A071242 Arithmetic mean of n and R(n) where n is a number such that the least significant digit and the most significant digits are of same parity and R(n) is its digit reversal (A004086).

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

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A071266 Smallest numbers which can be written in exactly n different ways as k + R(k) where R(k) is k reversed (A004086).

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A071687 Non-palindromic numbers such that either x=q1.Rev[x] or Rev[x]=q2.x, where R[x]=A004086[x] and q1 or q2 are integers not divisible by 10.

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A072032 a(n) = gcd(2^n, reverse(2^n)) = gcd(2^n, A004086(2^n)) = A055483(2^n).

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A074164 Smallest k such that R(k) > n*k, where R(k) is the digit reversal of k (A004086) (the reversal of 10 is taken to be 01 = 1, etc.).

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A074861 Iccanobirt sequence: a(n) = R(a(n-1)) + R(a(n-2)) + R(a(n-3)) where a(1)=a(2)=a(3)=1 and R(n) (A004086) is the reverse of n.

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A077337 Numbers k such that k and R(k) both are squarefree where R(n) (A004086) is the digit reversal of n.