A056964 -id:A056964 - cp's OEIS frontend

A072366 Numbers x such that x + reverse of x is a prime.

1, 10, 100, 116, 118, 140, 142, 146, 158, 166, 170, 172, 178, 182, 188, 190, 196, 215, 217, 229, 239, 241, 245, 257, 265, 269, 271, 277, 281, 287, 295, 299, 314, 316, 328, 338, 340, 344, 356, 364, 368, 370, 376, 380, 386, 394, 398, 413, 415, 427, 437, 439

Offset: 1

Shyam Sunder Gupta, Jul 18 2002

			116 is a term because 116+611=727 is a prime.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A056964, A072367.

Select[Range[1000], PrimeQ[ # + FromDigits[Reverse[IntegerDigits[ # ]]]] &] (* Tanya Khovanova, Jul 23 2007 *)

isok(n) = isprime(n+subst(Polrev(digits(n)), x, 10)); \\ Michel Marcus, Nov 29 2014

A055948 n + reversal of base 4 digits of n (written in base 10).

Original entry on oeis.org

0, 2, 4, 6, 5, 10, 15, 20, 10, 15, 20, 25, 15, 20, 25, 30, 17, 34, 51, 68, 25, 42, 59, 76, 33, 50, 67, 84, 41, 58, 75, 92, 34, 51, 68, 85, 42, 59, 76, 93, 50, 67, 84, 101, 58, 75, 92, 109, 51, 68, 85, 102, 59, 76, 93, 110, 67, 84, 101, 118, 75, 92, 109, 126, 65, 130, 195

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits in base 4 then a(n) is a multiple of 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Reverse and Add!

Cf. A030103, A056964.

Cf. A035524 (iterated).

a055948 n = n + a030103 n
-- Reinhard Zumkeller, Oct 10 2011

Table[n+FromDigits[Reverse[IntegerDigits[n,4]],4],{n,0,70}] (* Harvey P. Dale, Nov 24 2021 *)

a(n) = n + A030103(n).

A063433 'Reverse and Add!' trajectory of 10577.

Original entry on oeis.org

10577, 88078, 175166, 836737, 1574375, 7309126, 13528163, 49710694, 99312488, 187733887, 976071668, 1842242347, 9274664828, 17559329557, 93151725128, 175304440267, 937348843838, 1775697687577, 9533565653348, 17967131306707

Offset: 0

Klaus Brockhaus, Jul 20 2001

			a(1) = 10577 + 77501 = 88078.

Harry J. Smith, Table of n, a(n) for n=0,...,200
Index entries for sequences related to Reverse and Add!

A033865, A023108, A006960, A063434.

Cf. A056964, A004086.

m := 10577; 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;

a063433 n = a063433_list !! n
a063433_list = iterate a056964 10577 -- Reinhard Zumkeller, Sep 22 2011

NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&,10577, 20]  (* Harvey P. Dale, Apr 03 2011 *)

Rev(x)= { local(d,r); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ for (n=0, 200, if (n, a+=Rev(a), a=10577); write("b063433.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 21 2009

A345111 a(n) = n + A345110(n).

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 66, 77, 88

Offset: 0

Felix Fröhlich, Jun 09 2021

First differs from both A052008 and A056964 at n = 101.

			For n = 101: 101 + A345110(101) = 101 + 11 = 112, so a(101) = 112.

Cf. A052008, A056964, A345110, A345112, A345113, A345114, A345115.

Array[#+FromDigits@RotateLeft@IntegerDigits@#&,100,0] (* Giorgos Kalogeropoulos, Jun 09 2021 *)

eva(n) = subst(Pol(n), x, 10)
rot(vec) = if(#vec < 2, return(vec)); my(s=concat(Str(2), ".."), v=[]); s=concat(s, Str(#vec)); v=vecextract(vec, s); v=concat(v, vec[1]); v
a(n) = n + eva(rot(digits(n)))

def rotl(s): return s[1:] + s[0]
def a(n): return n + int(rotl(str(n)))
print([a(n) for n in range(63)]) # Michael S. Branicky, Jun 09 2021

A345114 Numbers whose trajectories under the map x -> A345111(x) do not reach a palindrome (conjectured).

Original entry on oeis.org

49, 58, 59, 67, 68, 69, 76, 77, 78, 79, 85, 86, 87, 88, 94, 95, 96, 97, 103, 114, 115, 116, 117, 119, 121, 124, 125, 126, 128, 129, 131, 134, 135, 137, 138, 139, 141, 142, 143, 146, 148, 149, 151, 153, 154, 155, 157, 158, 159, 160, 161, 162, 163, 164, 165, 168

Offset: 1

Felix Fröhlich, Jun 09 2021

The trajectories of the given terms do not reach a palindrome in 10000 (10^4) or fewer steps. The trajectory of 49 does not reach a palindrome in 100000 (10^5) or fewer steps.

Cf. A023108 (analog for the map x -> A056964(x)), A345110, A345111, A345112, A345113, A345115.

eva(n) = subst(Pol(n), x, 10)
rot(vec) = if(#vec < 2, return(vec)); my(s=concat(Str(2), ".."), v=[]); s=concat(s, Str(#vec)); v=vecextract(vec, s); v=concat(v, vec[1]); v
a345112(n, bound) = my(x=n, i=0); while(1, x=x+eva(rot(digits(x))); i++; if(digits(x)==Vecrev(digits(x)), break); if(i > bound, return(-1))); i
is(n) = a345112(n, 10000)==-1

def pal(s): return s == s[::-1]
def rotl(s): return s[1:] + s[0]
def A345111(n): return n + int(rotl(str(n)))
def A345112_bd(n, bd=10000):
    i, iter, seen = 0, n, set()
    while not (iter > n and pal(str(iter))) and iter not in seen and i < bd:
        seen.add(iter)
        i, iter = i+1, A345111(iter)
    return i if iter > n and pal(str(iter)) else 0
def aupto(lim, bd=10000):
    return [n for n in range(1, lim+1) if A345112_bd(n, bd=bd) == 0]
print(aupto(168, bd=100)) # Michael S. Branicky, Jun 09 2021

A349239 a(n) = n + (reversal of digits in the Zeckendorf representation of n).

Original entry on oeis.org

0, 2, 3, 4, 8, 6, 12, 11, 9, 18, 16, 15, 24, 14, 28, 24, 22, 36, 22, 36, 32, 22, 44, 37, 33, 55, 32, 54, 47, 33, 55, 48, 44, 66, 35, 70, 58, 51, 86, 48, 83, 71, 48, 83, 71, 64, 99, 51, 86, 74, 67, 102, 64, 99, 87, 56, 112, 92, 80, 136, 74, 130, 110, 72, 128, 108

Offset: 0

Kevin Ryde, Nov 11 2021

Kevin Ryde, Table of n, a(n) for n = 0..10000
Kevin Ryde, PARI/GP Code

Cf. A189920 (Zeckendorf digits), A349238 (reverse), A349240 (reverse and subtract), A348570 (Lychrels).

Other bases: A055944 (binary), A056964 (decimal).

PARI
```
\\ See links.
```

# Using functions NumToFib and RevFibToNum from A349238.
n, a = 0, 0
print(a + a, end = ", ")
while n < 65:
    n += 1
    print(n + RevFibToNum(NumToFib(n)), end = ", ") # A.H.M. Smeets, Nov 14 2021

a(n) = n + A349238(n).

a(n) = 2*n - A349240(n).

A063051 'Reverse and Add!' trajectory of 879.

Original entry on oeis.org

879, 1857, 9438, 17787, 96558, 182127, 903408, 1707717, 8884788, 17759676, 85455447, 159910905, 668930856, 1326970722, 3597766953, 7194444906, 13288889823, 46187778054, 91275556218, 172541113437, 906852258708

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 879 + 978 = 1857.

Harry J. Smith and Michael Lee, Table of n, a(n) for n = 0..2366 (Harry J. Smith provided terms 0 through 200)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063052, A056964, A004086.

m := 879; stop := 25; 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;

a033651 n = a033651_list !! n
a033651_list = iterate a056964 9 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 879, 40] (* Vincenzo Librandi, Sep 23 2013 *)

Rev(x)= { local(d); r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
{ for (n=0, 200, if (n, a+=Rev(a), a=879); write("b063051.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 16 2009

A063057 'Reverse and Add!' trajectory of 7059.

Original entry on oeis.org

7059, 16566, 83127, 155265, 717816, 1336533, 4692864, 9375828, 17661567, 94178238, 177465387, 961030158, 1812060327, 9042662508, 17095324917, 89037683988, 177976357086, 858730036857, 1617360074715, 6792060711876

Offset: 0

Klaus Brockhaus, Jul 07 2001

			a(1) = 7059 + 9507 = 16566.

Reinhard Zumkeller and Michael Lee, Table of n, a(n) for n = 0..2432 (terms 0 through 250 provided by Reinhard Zumkeller)
Index entries for sequences related to Reverse and Add!

Cf. A033865, A023108, A006960, A063058, A056964, A004086.

m := 7059; stop := 25; 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;

a063057 n = a063057_list !! n
a063057_list = iterate a056964 7059 -- Reinhard Zumkeller, Sep 22 2011

NestList[# + FromDigits[Reverse[IntegerDigits[#]]]&, 7059, 40] (* Vincenzo Librandi, May 03 2014 *)

A256756 a(n) = bitwise XOR of n and the reverse of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 25, 18, 39, 60, 45, 86, 67, 72, 22, 25, 0, 55, 50, 45, 36, 83, 78, 65, 29, 18, 55, 0, 9, 22, 27, 108, 117, 122, 44, 39, 50, 9, 0, 27, 110, 101, 100, 111, 55, 60, 45, 22, 27, 0, 121, 114, 111, 100, 58, 45, 36, 27, 110, 121

Offset: 0

Alois P. Heinz, Apr 09 2015

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

Cf. A003987, A004086, A055483, A056964, A056965, A061205, A068634, A175919, A256754, A256755.

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

Table[BitXor[n,FromDigits[Reverse[IntegerDigits[n]]]],{n,0,65}] (* Ivan N. Ianakiev, Apr 10 2015 *)

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

a(n) = A003987(n, A004086(n)).

A367796 Primes p such that the sum of p and its reversal is the square of a prime.

Original entry on oeis.org

2, 29, 47, 83, 20147, 23117, 24107, 63113, 80141, 81131, 261104399, 262005299, 262104299, 262203299, 263302199, 264203099, 264302099, 264500099, 270401489, 271500389, 273104189, 273302189, 274401089, 282203279, 284302079, 284500079, 291104369, 291203369, 292005269, 293005169, 293104169, 294302069

Offset: 1

Robert Israel, Nov 30 2023

Terms > 83 have an odd number of digits and an even first digit.

			A056964(a(n)) = 121 = 11^2 for 2 <= n <= 4.
A056964(a(n)) = 94249 = 307^2 for 5 <= n <= 10.
A056964(a(n)) = 1254505561 = 35419^2 for 11 <= n <= 71.

David A. Corneth, Table of n, a(n) for n = 1..5743 (terms <= 10^15)
David A. Corneth, PARI program

Cf. A056964, A067030, A061783. Subset of A367793.

digrev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(t) local v;
  v:= sqrt(t+digrev(t));
  v::integer and isprime(v)
end proc:
R:= 2, 29, 47, 83: count:= 4: flag:= true:
for d from 3 to 9 by 2 do
  p:= prevprime(10^(d-1));
  for i from 1 do
    p:= nextprime(p);
    p1:= floor(p/10^(d-1));
    if p1::odd then p:= nextprime((p1+1)*10^(d-1)) fi;
    if p > 10^d then break fi;
    if filter(p) then
       count:= count+1; R:= R,p;
fi od od:
R;

Select[Prime[Range[10^6]], PrimeQ[Sqrt[#+FromDigits[Reverse[IntegerDigits[#]]]]] &] (* Stefano Spezia, Dec 10 2023 *)

PARI
```
\\ See PARI link
```

cp's OEIS Frontend

A072366 Numbers x such that x + reverse of x is a prime.

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PARI

A055948 n + reversal of base 4 digits of n (written in base 10).

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A063433 'Reverse and Add!' trajectory of 10577.

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A345111 a(n) = n + A345110(n).

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A345114 Numbers whose trajectories under the map x -> A345111(x) do not reach a palindrome (conjectured).

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PARI

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A349239 a(n) = n + (reversal of digits in the Zeckendorf representation of n).

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A063051 'Reverse and Add!' trajectory of 879.

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A063057 'Reverse and Add!' trajectory of 7059.

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