A004086 -id:A004086 - cp's OEIS frontend

A070246 a(n) = lcm(n, R(n)) / gcd(n, R(n)), where R(n) (A004086) is the digit reversal of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 28, 403, 574, 85, 976, 1207, 18, 1729, 10, 28, 1, 736, 28, 1300, 403, 24, 574, 2668, 10, 403, 736, 1, 1462, 1855, 28, 2701, 3154, 403, 10, 574, 28, 1462, 1, 30, 736, 3478, 28, 4606, 10, 85, 1300, 1855, 30, 1, 3640, 475, 4930, 5605

Offset: 1

Amarnath Murthy, May 09 2002

a(1) = 1, a(18) = 18. Are there more terms for which a(k) = k?

			a(12) = lcm(12,21)/gcd(12,21) = 84/3 = 28.

r[n_] := FromDigits[ Reverse[ IntegerDigits[n]]]; Table[ LCM[n, r[n]] / GCD[n, r[n]], {n, 1, 65}]

Edited by Robert G. Wilson v, May 10 2002

A070839 Smallest number k such that k + R(k) is a multiple of 11 different from all previous cases, where R(k) is digit reversal of k (A004086).

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 29, 39, 49, 59, 69, 79, 89, 99, 121, 132, 143, 154, 165, 176, 187, 198, 209, 319, 429, 539, 649, 759, 869, 979, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018

Offset: 1

Amarnath Murthy, May 12 2002

Corrected and extended by Larry Reeves (larryr(AT)xnet.com), Oct 04 2002

A072033 Smallest x > 0 such that gcd(2^x, A004086(2^x)) = 2^n.

Original entry on oeis.org

4, 1, 2, 3, 26, 131, 227, 301, 567, 879, 3240, 11051, 8048, 38911, 7321, 97309, 108190, 6294, 138124, 4675268, 2687104, 1336154, 5774420

Offset: 1

Labos Elemer, Jun 07 2002

a(14)=7321, a(17)=6294.

			n=4: a(4)=26 because gcd(2^26, reverse(2^26)) = gcd(67108864, 46880176) = 16 = 2^n.

Cf. A055483, A004086, A071686, A072032.

a[n_] := Block[{k=1}, While[ IntegerExponent[ GCD[2^k, FromDigits@ Reverse@ IntegerDigits[2^k]], 2] != n, k++]; k]; Array[a, 13, 0] (* Giovanni Resta, Oct 28 2019 *)

a(n) = min{x: gcd(2^x, reverse(2^x))=2^n} = min{x: A055483(x)=2^n}.

A072032(a(n)) = 2^n.

Offset corrected, missing a(3) and a(13)-a(22) added by Giovanni Resta, Oct 28 2019

A088490 a(n) is the absolute value of p minus A004086(p), where (p-2,p) is the n-th pair of twin primes.

Original entry on oeis.org

0, 0, 18, 72, 18, 9, 45, 36, 198, 792, 792, 0, 0, 198, 792, 693, 99, 99, 99, 0, 594, 297, 99, 99, 198, 396, 495, 297, 297, 495, 693, 495, 99, 99, 495, 180, 2268, 450, 2538, 2808, 2358, 90, 8442, 630, 1728, 90, 7812, 2088, 2358, 8352, 7452, 360

Offset: 1

Cino Hilliard, Nov 09 2003

9 divides each term.

			The 4th pair of twin primes is (17,19). a(4) then is the absolute value of 19 - 91, which is 72.

a = {}; For[n = 1, n < 275, n++, If[Prime[n + 1] == Prime[n] + 2, AppendTo[a, Abs[Prime[n + 1] - FromDigits[Reverse[IntegerDigits[Prime[n + 1]]]]]]]]; a

revdifftp2(n) = { forprime(x=1,n, if(isprime(x+2), a=vector(x); x1=x+2; z=0; ln=length(Str(x1)); for(y=1,ln, a[y] = x1%10; x1=floor(x1/10); ); for(y=1,ln, z += a[y]*10^(ln-y); ); print1(abs(z-(x+2))" "); ) ) }

Edited by Stefan Steinerberger, Jul 22 2007

A178836 Numbers n such that the period of 1/n equals the period of 1/R(n), where R(n) (A004086) is the reversal of n.

Original entry on oeis.org

3, 7, 9, 11, 33, 77, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 707, 717, 727, 737, 747, 757, 767, 777, 787, 797, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999, 1001, 1111, 1221, 1331, 1441, 1551

Offset: 1

Michel Lagneau, Jun 16 2010

Non-palindromic numbers are included in this sequence :

{3267, 3927, 7293, 7632,...}

			3267 is in the sequence because period (1/3267) = 66 and also period(1/7623) = 66.
3927 is in the sequence because period (1/3927) = 48 and also period(1/7293) = 48.

Cf. A004086, A045572, A002329.

with(numtheory): nn:=8000:for n from 3 to nn do: s:=0:l:=length(n):for q from 0 to l-1 do:x:=iquo(n,10^q):y:=irem(x,10):s:=s+y*10^(l-1-q): od: indic1:=0:for p from 1 to nn do:if irem(10^p, n) = 1 and gcd(n, 5) = 1 and indic1=0 then pp:=p: indic1:=1:else fi:od: indic2:=0:for p from 1 to nn do:if irem(10^p, s) = 1 and gcd(s, 5) = 1 and indic2=0 then ppp:=p:indic2:=1:else fi:od: if pp=ppp and indic1=1 and indic2=1 then print(n):else fi:od:

A289140 Positive numbers k such that rev(k)^2 + rev(k^2) is a square, where rev(n) = A004086(n) is the digital reverse of n.

Original entry on oeis.org

998586, 3632658, 9985860, 36326580, 74471091, 99664458, 99858600, 363265800, 634826115, 743193501, 744710910, 756335085, 759317343, 996644580, 998586000, 3632658000, 6348261150, 7177621788, 7431935010, 7447109100, 7563350850, 7593173430, 9966445800

Offset: 1

Giovanni Resta, Jun 26 2017

Every term must be a multiple of 3.

			998586 is a term since rev(998586^2) + 685899^2 = 1079100^2.

Cf. A004086, A061230, A061231, A068536, A102859, A113798, A202386, A256515.

rev[n_] := FromDigits@ Reverse@ IntegerDigits@ n; Parallelize@ Select[3 Range[4 10^6], IntegerQ@ Sqrt[rev[#^2] + rev[#]^2] &]

isok(n) = issquare(fromdigits(Vecrev(digits(n)))^2 + fromdigits(Vecrev(digits(n^2)))); \\ Michel Marcus, Jun 29 2017

A307799 a(0) = 0, a(1) = 3; a(n) = rev(a(n-1))*a(n-1) + a(n-2), where rev = digit reversal (A004086).

Original entry on oeis.org

0, 3, 9, 84, 4041, 5673648, 48020423368761, 806086788756824484462571572, 221815145293562950532110825781341443907408910699844537

Offset: 0

Ilya Gutkovskiy, Apr 29 2019

The next term is too large to include.

			+---+--------------+---------------------+------------------+
| n | a(n)/a(n+1)  | Continued fraction  |      Comment     |
+---+--------------+---------------------+------------------+
| 1 |    3/9       | [0; 3]              |    3 = rev(a(1)) |
+---+--------------+---------------------+------------------+
| 2 |    9/84      | [0; 9, 3]           |    9 = rev(a(2)) |
+---+--------------+---------------------+------------------+
| 3 |   84/4041    | [0; 48, 9, 3]       |   48 = rev(a(3)) |
+---+--------------+---------------------+------------------+
| 4 | 4041/5673648 | [0; 1404, 48, 9, 3] | 1404 = rev(a(4)) |
+---+--------------+---------------------+------------------+

Cf. A001040, A004086, A058182.

a[n_] := a[n] = FromDigits[Reverse[IntegerDigits[a[n - 1]]]] a[n - 1] + a[n - 2]; a[0] = 0; a[1] = 3; Table[a[n], {n, 0, 8}]

A332850 Numbers k = a^2 + b^2 such that reversal(k) = a^2 - b^2 for a > b > 0, where reversal is A004086.

Original entry on oeis.org

699796, 4854634, 6752626, 84036010, 931910661, 21584860960, 52554850525, 467170024564, 637843128736, 638730439636, 638734039636, 638943127636, 727830438745, 727834038745, 746710459825, 746754019825, 748943127625, 9894192267061, 401309596403104, 844181015028970

Offset: 1

Metin Sariyar, Feb 26 2020

When b=0, the palindromic numbers m = a^2 + b^2 such that reversal(m) = a^2 - b^2, are A002779 (palindromic squares).

a(19) > 3*10^14, if it exists. - Giovanni Resta, Feb 27 2020

			699796 = 836^2 + 30^2 and 697996 = 836^2 - 30^2.

Cf. A002779, A004086, A035519, A055096, A202386.

Do[If[IntegerReverse[a^2+b^2]==a^2-b^2,Print[{a^2+b^2,a,b}]],{a,1,50000},{b,1,a-1}]

isok(k) = {my(r = fromdigits(Vecrev(digits(k))), s = r+k, d = k-r); d && !(s % 2) && issquare(s/2) && !(d % 2) && issquare(d/2); } \\ Michel Marcus, Feb 27 2020

a(6)-a(18) from Giovanni Resta, Feb 27 2020

a(19)-a(20) from Jinyuan Wang, Apr 10 2025

A369612 Numbers k such that k divides Sum_{i=1..k} A004086(i).

Original entry on oeis.org

1, 3, 5, 7, 9, 18, 50, 89, 147, 161, 702, 999, 1323, 1998, 2091, 3042, 3072, 3753, 7982, 13408, 32493, 33867, 99999, 179922, 199998, 368121, 375897, 384876, 741137, 993006, 1173628, 1410462, 2021017, 3160593, 7212528, 9975778, 9999999, 17052657, 18093882, 18782433

Offset: 1

Ctibor O. Zizka, Jan 27 2024

Numbers k that divide A062918(k).

			Sum_{i=1..18} A004086(i) / 18 = 414/18 = 23 thus k = 18 is a term.

Cf. A004086, A062918.

With[{m = 10^7}, Position[Accumulate[Array[IntegerReverse, m]]/Range[m], ?IntegerQ] // Flatten] (* _Amiram Eldar, Jan 27 2024 *)

A370842 Numbers k that can be added without carries to their digit reversal (A004086(k)).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 60, 61, 62, 63, 70, 71, 72, 80, 81, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113

Offset: 1

Rémy Sigrist, Mar 03 2024

All positive terms belong to A015976.

			42 belongs to the sequence as 42 + 24 does not lead to carries.
48 does not belong to the sequence as 48 + 84 leads to carries.

Cf. A004086, A015976, A056964, A140900 (base-2 analog).

is(n, base = 10) = { my (d = if (n, digits(n, base), [0]), p = d + Vecrev(d)); vecmax(p) < base }

cp's OEIS Frontend

A070246 a(n) = lcm(n, R(n)) / gcd(n, R(n)), where R(n) (A004086) is the digit reversal of n.

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A070839 Smallest number k such that k + R(k) is a multiple of 11 different from all previous cases, where R(k) is digit reversal of k (A004086).

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A072033 Smallest x > 0 such that gcd(2^x, A004086(2^x)) = 2^n.

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A088490 a(n) is the absolute value of p minus A004086(p), where (p-2,p) is the n-th pair of twin primes.

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A178836 Numbers n such that the period of 1/n equals the period of 1/R(n), where R(n) (A004086) is the reversal of n.

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A289140 Positive numbers k such that rev(k)^2 + rev(k^2) is a square, where rev(n) = A004086(n) is the digital reverse of n.

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A307799 a(0) = 0, a(1) = 3; a(n) = rev(a(n-1))*a(n-1) + a(n-2), where rev = digit reversal (A004086).

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A332850 Numbers k = a^2 + b^2 such that reversal(k) = a^2 - b^2 for a > b > 0, where reversal is A004086.

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A369612 Numbers k such that k divides Sum_{i=1..k} A004086(i).

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