A056964 -id:A056964 - cp's OEIS frontend

A066054 'Reverse and Add!' trajectory of 10583.

10583, 49084, 97178, 184357, 937838, 1776577, 9533348, 17966707, 88733678, 176367466, 841131137, 1572262285, 7394885036, 13700769973, 51697470704, 92404950319, 183710890748, 1030808908129, 10248906988430, 13737867972631

Offset: 0

Klaus Brockhaus, Nov 30 2001

			a(1) = 10583 + 38501 = 49084.

Harry J. Smith and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (the first 150 terms from Harry J. Smith)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A066055.

Cf. A056964, A004086.

: m := 10583; stop := 20; c := 0; rev := int_reverse(m); while m <> rev and c < stop do inc(c); write(m," "); m := m + rev; rev := int_reverse(m); end;

a066054 n = a066054_list !! n
a066054_list = iterate a056964 10583 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&,  10583, 40] (* Vincenzo Librandi, May 03 2014 *)

Rev(x)= { local(d, r=0); while (x>0, d=x%10; x\=10; r=r*10 + d); return(r) } { a=10583; for (n = 0, 150, if (n, a+=Rev(a)); write("b066054.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 08 2009

A256754 a(n) = bitwise AND of n and the reverse of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 4, 13, 8, 3, 16, 1, 16, 19, 0, 4, 22, 0, 8, 16, 26, 8, 16, 28, 2, 13, 0, 33, 34, 33, 36, 1, 2, 5, 0, 8, 8, 34, 44, 36, 0, 10, 16, 16, 0, 3, 16, 33, 36, 55, 0, 9, 16, 27, 4, 16, 26, 36, 0, 0, 66, 64, 68, 64, 6, 1, 8, 1, 10

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A004086, A004198, A055483, A056964, A056965, A061205, A068634, A256755, A256756.

a:= n-> Bits[And](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitAnd[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,74}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitand(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A004198(n,A004086(n)).

A256755 a(n) = bitwise OR of n and the reverse of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 29, 31, 47, 63, 61, 87, 83, 91, 22, 29, 22, 55, 58, 61, 62, 91, 94, 93, 31, 31, 55, 33, 43, 55, 63, 109, 119, 127, 44, 47, 58, 43, 44, 63, 110, 111, 116, 127, 55, 63, 61, 55, 63, 55, 121, 123, 127, 127, 62, 61, 62, 63, 110

Offset: 0

Alois P. Heinz, Apr 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Bitwise operation

Cf. A003986, A004086, A055483, A056964, A056965, A061205, A068634, A256754, A256756.

a:= n-> Bits[Or](n, (s-> parse(cat(s[-i]$i=1..length(s))))(""||n)):
seq(a(n), n=0..80);

Table[BitOr[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,64}] (* Ivan N. Ianakiev, Apr 10 2015 *)

a(n) = bitor(n, subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Apr 10 2015

a(n) = A003986(n,A004086(n)).

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

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

Square roots of the squares in A067030.

			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.

Hugo Pfoertner, Table of n, a(n) for n = 1..2253, using results from participants Sebastian and l4m2 at the Code Golf challenge.
Nicolay Avilov, Problem 2422. Mirror numbers (in Russian).
Code Golf Stack Exchange, RADD decomposition of an integer, coding challenge started Jan 01 2023.

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

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.

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

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

A055954 a(n) = n + reversal of base 7 digits of n (written in base 10).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 8, 16, 24, 32, 40, 48, 56, 16, 24, 32, 40, 48, 56, 64, 24, 32, 40, 48, 56, 64, 72, 32, 40, 48, 56, 64, 72, 80, 40, 48, 56, 64, 72, 80, 88, 48, 56, 64, 72, 80, 88, 96, 50, 100, 150, 200, 250, 300, 350, 64, 114, 164, 214, 264, 314, 364, 78, 128, 178

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits in base 7 then a(n) is a multiple of 8

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A030106, A056964.

Table[n+FromDigits[Reverse[IntegerDigits[n,7]],7],{n,0,70}] (* Harvey P. Dale, Jan 29 2020 *)

a(n) = n + fromdigits(Vecrev(digits(n, 7)), 7); \\ Michel Marcus, Aug 08 2024

a(n) = n + A030106(n).

A072367 Squares x such that x + reverse of x is a prime.

Original entry on oeis.org

1, 100, 196, 625, 10816, 13456, 15376, 18496, 21025, 22201, 22801, 24649, 25921, 27889, 29929, 33856, 35344, 36100, 42025, 50176, 60025, 63001, 70756, 71824, 73984, 78400, 82369, 83521, 96100, 1012036, 1048576, 1073296, 1123600, 1144900

Offset: 1

Shyam Sunder Gupta, Jul 18 2002

			196 is a term because 196 is a square and 196+691=887 is a prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A056964, A072366.

[k:k in [m^2:m in [1..1100]]| IsPrime(Seqint(Intseq(k,10))+Seqint(Reverse(Intseq(k,10))))]; // Marius A. Burtea, Jun 20 2019

select(t -> isprime(t+revdigs(t)), [seq(i^2,i=1..10000)]); # Robert Israel, Jun 19 2019

Select[Range[1100]^2,PrimeQ[#+FromDigits[Reverse[IntegerDigits[#]]]]&]  (* Harvey P. Dale, Feb 20 2011 *)

isok(x) = issquare(x) && isprime(x+fromdigits(Vecrev(digits(x)))); \\ Michel Marcus, Jun 19 2019

A072385 Primes which can be represented as the sum of a prime and its reverse.

Original entry on oeis.org

383, 443, 463, 787, 827, 887, 929, 1009, 1049, 1069, 1151, 1171, 1231, 1373, 1453, 1493, 1777, 30203, 30403, 31013, 32213, 32413, 32423, 33023, 33223, 34033, 34843, 35053, 36263, 36653, 37273, 37463, 37663, 38083, 38273, 38873, 39293, 39883

Offset: 1

Shyam Sunder Gupta, Jul 20 2002

This set is the image under the "reverse and add" operation (A056964) of the Luhn primes A061783 (which remain prime under that operation). Those have always an odd number of digits, and start with an even digit. Therefore this sequence has its terms in intervals [3,20]*100^k with k = 1, 2, 3.... - M. F. Hasler, Sep 26 2019

			383 is a term because it is prime and it is the sum of prime 241 and its reverse 142.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..253 from M. F. Hasler)
Matt Parker, 383 is cool, Numberphile series on YouTube, Feb. 15, 2017.

Cf. A004086 (reverse), A004087 (primes reversed), A056964 (reverse & add), A061783 (Luhn primes), A086002 (similar, using "rotate" instead of "reverse").

f@n_:=(Select[# + IntegerReverse[#] & /@ Prime[Range[n]], PrimeQ@# && # <= Prime[n] &] // Union); f@3000 (* Harvey P. Dale, Jul 18 2018; corrected by Hans Rudolf Widmer, Aug 15 2024 *)

is_A072385(p)={isprime(p)&&forprime(q=p\10,p*9\10,A056964(q)==p&&return(1))} \\ A056964(n)=n+fromdigits(Vecrev(digits(n))). It is much faster to produce the terms as shown below, rather than to "select" them from a range of primes. - M. F. Hasler, Sep 26 2019

A072385=Set(apply(A056964, A061783)) \\ with, e.g.: A061783=select(is_A061783(p)={isprime(A056964(p))&&isprime(p)}, primes(8713)) - M. F. Hasler, Sep 26 2019

a(n) = A056964(A061783(n)). - M. F. Hasler, Sep 26 2019

Cross-references added by M. F. Hasler, Sep 26 2019

A190481 Number of distinct integers with n digits which are the image of integers by the function Reverse and Add!.

Original entry on oeis.org

4, 14, 93, 256, 1793, 4872, 34107, 92590, 648154, 1759313, 12315269, 33427272, 233991155, 635119194, 4445835138, 12067267861, 84470877438, 229278099157, 1604946701532, 4356283914175, 30493987422124, 82769394462323, 579385761306789, 1572618495070552

Offset: 1

Aldo González Lorenzo, May 25 2011

a(n) is the cardinality of the set of Image(Reverse and Add!) intersected with [10^(n-1), 10^n[. Here we suppose that the domain of the function Reverse and Add! is {1, 2, 3, ...}

There are 4, 50, 450, 4590, 45405,... (A232731) ways to obtain integers with n = 1,2,... digits as images under the function "Reverse and add!", but many result in the same image and are counted here only once. Example: 11+digrev(11) = 22 and 20+digrev(20)=22 contribute only once to the set of distinct images at n=2. - R. J. Mathar, Jun 17 2011

			Example: let RaA(x) be the function Reverse and Add!, then:
RaA(1)=2
RaA(2)=4
RaA(3)=6
RaA(4)=8
RaA(5)=10
RaA(6)=11, ...
So a(1) is the cardinal of {2,4,6,8}, which is 4:

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

Cf. A056964, A067030.

A055642 := proc(n) max(1,1+ilog10(n)) ; end proc:
A056964 := proc(n) n+digrev(n) ; end proc:
A190481 := proc(n) local s,i,ra ; s := {} ; for i from 1 to 10^n do ra := A056964(i) ; if A055642( ra) = n then s := s union {ra}  ; end if; end do: nops(s) ; end proc:
for n from 1 do print(n,A190481(n)) ; end do: # R. J. Mathar, Jun 17 2011

Empirical g.f.: x*(4 + 18*x + 23*x^2 - 29*x^3 - 58*x^4 - 34*x^5 - 81*x^6 - 45*x^7 - 32*x^8 - 9*x^9) / ((1 + x)*(1 - 19*x^2)*(1 - 2*x + x^2 - x^3)*(1 + 2*x + x^2 + x^3)). - Colin Barker, Mar 20 2017

a(9)-a(10) from Lars Blomberg, Dec 01 2013

a(11)-a(24) from Hiroaki Yamanouchi, Sep 04 2014

A256398 Palindromes of the form i^2 + reverse(i)^2.

Original entry on oeis.org

0, 2, 8, 101, 242, 404, 585, 909, 10001, 12221, 14841, 20402, 24642, 40004, 44244, 48884, 50805, 90009, 96269, 1000001, 1030301, 1080801, 1210121, 1244421, 1298921, 1440441, 1478741, 1690961, 2004002, 2234322, 2468642, 2484842, 4000004, 4050504, 4410144

Offset: 1

Bui Quang Tuan, Mar 28 2015

Is 864666666468 the only term in this sequence that has an even number of digits? - Jon E. Schoenfield, Mar 30 2015

The next terms with an even number of digits are 5807785995877085, 56359464311346495365, and 943614966934439669416349, which are obtained for i = 37939066, 3553782166, 529145826418 (and their reverses). - Giovanni Resta, Aug 22 2025

			Palindrome 585 is in the sequence because 585 = 12^2 + 21^2.
The smallest term that can be obtained in more than one way is 125484521 = 11020^2 + 2011^2 = 11200^2 + 211^2. Are there any terms that can be obtained in more than two ways? - _Jon E. Schoenfield_, Mar 30 2015

Bui Quang Tuan, Table of n, a(n) for n = 1..458

Cf. A002113 (palindromes), A056964 (n+rev(n)).

Cf. A256437.

Sort@ DeleteDuplicates@ Select[Table[n^2 + FromDigits[Reverse[IntegerDigits@ n]]^2, {n, 10000}], Reverse@ IntegerDigits@ # == IntegerDigits@ # &] (* Michael De Vlieger, Mar 28 2015 *)

rev(n)=r="";d=digits(n);for(i=1,#d,r=concat(Str(d[i]),r));eval(r)
v=[];for(n=0,10^4,if(rev(P=(n^2+rev(n)^2))==P,v=concat(v,P)));vecsort(v,,8) \\ Derek Orr, Mar 29 2015

Data corrected by Derek Orr, Mar 29 2015

A319603 a(n) = n^3 + reversal of digits of n^3.

Original entry on oeis.org

0, 2, 16, 99, 110, 646, 828, 686, 727, 1656, 1001, 2662, 9999, 10109, 7216, 9108, 11000, 8107, 8217, 16445, 8008, 10890, 95249, 88288, 56655, 68276, 85147, 58374, 47864, 122731, 27072, 49583, 119491, 109890, 79697, 100699, 112320, 86258, 82717, 150714, 64046, 81907, 162135, 150104

Offset: 0

Seiichi Manyama, Sep 24 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

n^b + reversal of digits of n^b: A056964 (b=1), A061226 (b=2), this sequence (b=3).

Cf. A000578, A004165, A072384 (subsequence of primes).

{a(n) = n^3+fromdigits(Vecrev(digits(n^3)))}

a(n) = A000578(n) + A004165(n).

cp's OEIS Frontend

A066054 'Reverse and Add!' trajectory of 10583.

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A256754 a(n) = bitwise AND of n and the reverse of n.

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A256755 a(n) = bitwise OR of n and the reverse of n.

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

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A055954 a(n) = n + reversal of base 7 digits of n (written in base 10).

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A072367 Squares x such that x + reverse of x is a prime.

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A072385 Primes which can be represented as the sum of a prime and its reverse.

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