A061783 -id:A061783 - cp's OEIS frontend

A086002 Primes which when added to their own rotation yield a prime.

229, 239, 241, 257, 269, 271, 277, 281, 439, 443, 463, 467, 479, 499, 613, 641, 653, 661, 673, 677, 683, 691, 811, 823, 839, 863, 881, 10111, 10151, 10169, 10181, 10243, 10247, 10253, 10267, 10303, 10313, 10331, 10343, 10391, 10429, 10453, 10457

Offset: 1

Chuck Seggelin, Jul 07 2003

Let rotation rot(k) of a number k be defined by swapping the blocks of the first [d/2] and of the last [d/2] digits of k, where d=A055642(k). If the number of digits in k is odd, the center digit remains untouched during rotation.
So for example the rotation of 1234 is 3412, while the rotation of 12345 is 45312.
Rotation differs from reversal (A004086) for numbers with at least 4 digits, that is, after A004087(168) if we are concerned with primes.
The sequence lists primes p such that p+rot(p) is (again) prime.
Differs from A061783, where rot(k) is replaced by reverse(k), from the 5-digit terms on. - M. F. Hasler, Mar 03 2011
a(n) has an odd number of digits (see RJM comment in A086004). If a(n) has 2m+1 digits, then the m-th digit of a(n) is even as otherwise a(n) + rot(a(n)) is even. - Chai Wah Wu, Aug 19 2015

			a(100)=12917 because (i) 12917 is prime and (ii) rotate(12917) = 17912 and 12917+17912=30829, which is also prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A086003, A086004.

A055642 := proc(n) max(1,1+ilog10(n)) ; end:
rot := proc(n) local d,dl,dh,pre,suf ; d := A055642(n) ; dl := floor( d/2) ; dh := floor( (d+1)/2) ; pre := floor(n/10^dh) ; suf := n mod 10^dl ; if dl <> dh then suf*10^dh+pre+10^dl*( floor(n/10^dl) mod 10) ; else suf*10^dh+pre ; fi; end:
isA086002 := proc(p) if isprime(p) then isprime(p+rot(p)) ; else false; fi; end:
for n from 1 to 1500 do p := ithprime(n) ; if isA086002(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, May 27 2009

rot[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];If[OddQ[len],FromDigits[ Join[Take[idn,-Floor[len/2]],{idn[[(len+1)/2]]},Take[idn,Floor[len/2]]]],FromDigits[ Join[ Take[ idn,-len/2],Take[idn,len/2]]]]]; Select[Prime[Range[1500]],PrimeQ[ #+rot[#]]&] (* Harvey P. Dale, Apr 26 2022 *)

Edited by R. J. Mathar, May 27 2009

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
```

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

A232446 Primes p such that reversal( p^2 ) + p is also prime.

Original entry on oeis.org

7, 151, 787, 1549, 1579, 2029, 2083, 2179, 2833, 2971, 4549, 4591, 4801, 4999, 5077, 5167, 5179, 5209, 5227, 5407, 6343, 6529, 6547, 6553, 6577, 6679, 7027, 7753, 7867, 7873, 7927, 7963, 7993, 8167, 8191, 8311, 9091, 9103, 9151, 9283, 14251, 14281, 14389, 14437

Offset: 1

K. D. Bajpai, Nov 24 2013

nonn

			a(1)= 7, it is prime: prime(4)= 7: reversal(7^2)+7= reversal(49)+7= 94+7= 101 which is also prime.
a(2)= 151, it is prime: prime(36)= 151: reversal(151^2)+151= reversal(22801)+151=10822+151= 10973 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..4250

Cf. A061783 (primes p: p+(p reversed) is also prime).

Function reversal is given by A004086. Cf. also A004087.

with(StringTools): KD:= proc() local a,p; p:=ithprime(n);a:= parse(Reverse(convert((p^2), string)))+p; if isprime(a) then RETURN (p): fi; end: seq(KD(), n=1..3000);

Select[Prime[Range[3000]], PrimeQ[# + FromDigits[Reverse[IntegerDigits[#^2]]]] &]

A304390 Prime numbers p such that p squared + (p reversed) squared is also prime.

Original entry on oeis.org

23, 41, 227, 233, 283, 401, 409, 419, 421, 461, 491, 499, 823, 827, 857, 877, 2003, 2083, 2267, 2437, 2557, 2593, 2617, 2633, 2677, 2857, 2887, 2957, 4001, 4021, 4051, 4079, 4129, 4211, 4231, 4391, 4409, 4451, 4481, 4519, 4591, 4621, 4639, 4651, 4871, 6091, 6301, 6329, 6379, 6521, 6529, 6551

Offset: 1

Pierandrea Formusa, Aug 16 2018

			The prime number 227 belongs to this sequence as 722 is 227 reversed and 227^2 + 722^2 = 572813, which is prime.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A061783 (Luhn primes).

Subsequence of A069207. - Michel Marcus, Aug 21 2018

Select[Prime@ Range@ 850, PrimeQ[#^2 + FromDigits[ Reverse@ IntegerDigits@ #]^2] &] (* Giovanni Resta, Sep 03 2018 *)

isok(p) = isprime(p) && isprime(p^2+eval(fromdigits(Vecrev(digits(p))))^2); \\ Michel Marcus, Aug 21 2018

nmax=10000
def is_prime(num):
    if num == 0 or num == 1: return(0)
    for k in range(2, num):
       if (num % k) == 0:
           return(0)
    return(1)
ris = ""
for i in range(nmax):
    r=int((str(i)[::-1]))
    t=pow(i,2)+pow(r,2)
    if is_prime(i):
       if is_prime(t):
          ris = ris+str(i)+","
print(ris)

A367793 Primes p such that the sum of p and its reversal is a semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 131, 151, 181, 191, 211, 223, 227, 233, 251, 293, 313, 353, 373, 383, 401, 409, 419, 421, 431, 433, 449, 457, 487, 491, 571, 599, 601, 607, 617, 619, 631, 643, 647, 727, 757, 787, 797, 809, 821, 827, 829, 853, 859, 877, 883, 919, 929, 2011

Offset: 1

Zak Seidov and Robert Israel, Nov 30 2023

Terms > 11 with an even number of digits have an even first digit.

			a(6) = 23 is a term because 23 is a prime and 23 + 32 = 55 = 5 * 11 is a semiprime.

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

Cf. A001358, A056964, A061783.

digrev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
select(p -> isprime(p) and numtheory:-bigomega(p+digrev(p))=2, [2,seq(i,i=3..10000,2)]);

Select[Prime[Range[10^3]], 2 == PrimeOmega[# + FromDigits[Reverse[IntegerDigits[#]]]] &]

A174402 Primes such that applying "reverse and add" twice produces two more primes.

Original entry on oeis.org

271, 281, 21491, 21991, 22091, 22481, 23081, 23971, 24071, 25951, 26681, 26981, 27271, 27431, 27691, 27791, 28031, 28661, 28921, 28961, 29021, 29191, 29251, 29411, 29671, 2129891, 2131991, 2141791, 2141891, 2151791, 2157091, 2161591, 2179391, 2191291

Offset: 1

Harvey P. Dale, Nov 27 2010

Some observations:
For all terms, the first digit is 2, last digit is 1, number of digits is odd: 3,5,7,...
The sequence is infinite. Number of 3-digit terms is 2, number of 5-digit terms is 23, number of 7-digit terms is 585, number of 9-digit terms is 26611.
Applying "reverse and add" a third time always produces composites. - Zak Seidov, Dec 09 2013

			21491 is included because (1) it is prime, and (2) 21491 + 19412 = 40903 which is prime, and (3) 40903 + 30904 = 71807 which also is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..610 (All terms with 7 or fewer digits.)

Cf. A061783.

Transpose[Select[Table[{Prime[i],And@@PrimeQ/@NestList[#+FromDigits[ Reverse[ IntegerDigits[#]]]&,Prime[i],2]},{i,500000}],#[[2]] == True&]][[1]]
tmpQ[p_]:=AllTrue[Rest[NestList[#+IntegerReverse[#]&,p,2]],PrimeQ]; Select[Prime[Range[163000]],tmpQ] (* Harvey P. Dale, Feb 05 2025 *)

A232655 Primes p such that reversal (p^2+p) is also prime.

Original entry on oeis.org

5, 13, 19, 103, 139, 181, 193, 271, 277, 313, 379, 433, 577, 619, 631, 853, 859, 883, 1093, 1117, 1123, 1237, 1279, 1321, 1741, 1873, 1933, 1987, 2659, 2707, 2713, 2719, 2767, 2791, 3163, 3217, 3271, 3331, 3469, 3529, 3547, 3631, 3637, 3727, 3907, 3943, 4129, 4177

Offset: 1

K. D. Bajpai, Nov 27 2013

			a(2)= 13, it is prime: n= 6, prime(6)= 13: reversal(13^2+13)= 281, which is also prime.
a(4)= 103, it is prime: n= 27, prime(27)= 103: reversal(103^2+103)= 21701, which is also prime.
a(6)= 181, it is prime: n= 42, prime(42)= 181: reversal(181^2+181)= 24923, which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..6650

Cf. A004087 (primes written backwards).
Cf. A061783 (primes p: p+(p reversed)is also prime).
Cf. A232446 (primes p: reversal(p^2)+p is also prime).

with(StringTools): KD:= proc() local a, p; p:=ithprime(n); a:= parse(Reverse(convert((p^2+p), string))); if isprime(a) then RETURN (p): fi; end: seq(KD(), n=1..3000);

Select[Prime[Range[3000]], PrimeQ[FromDigits[Reverse[IntegerDigits[#^2 + #]]]] &]

A244392 Primes p such that p + (p reversed) is a palindrome.

Original entry on oeis.org

2, 3, 11, 13, 17, 23, 29, 31, 41, 43, 47, 53, 61, 71, 83, 101, 103, 107, 113, 127, 131, 137, 211, 223, 227, 233, 241, 311, 313, 331, 401, 421, 431, 433, 443, 503, 521, 523, 541, 601, 613, 631, 641, 643, 701, 811, 821, 1013, 1021, 1031, 1033, 1051, 1061, 1063

Offset: 1

Vincenzo Librandi, Jul 02 2014

Palindrome is also a prime for n = 241, 443, 613, 641, 811, 20011, 20047, 20051, 20101, 20161, ... . Example: 613+316 = 929, which is prime. [Bruno Berselli, Jul 05 2014]

Subsequence of primes within A015976. - Michel Marcus, Jul 05 2014

			13 is in the sequence because 13+31 = 44 is a palindrome.
1103 is in the sequence because 1103+3011 = 4114 is a palindrome.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A007500, A015976, A061783.

[p: p in PrimesUpTo(1200) | q eq Reverse(q) where q is Intseq(p+Seqint(Reverse(Intseq(p))))]; // Bruno Berselli, Jul 05 2014

selQ[p_] := (id = IntegerDigits[p]; id2 = IntegerDigits[p + FromDigits[Reverse[id]]]; id2 == Reverse[id2]); Select[Array[Prime, 200], selQ] (* Jean-François Alcover, Jul 05 2014 *)
Select[Prime[Range[200]],PalindromeQ[#+IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 11 2020 *)

A309429 Least Luhn prime in base 2n: primes p such that p + reverse(p) in base 2n is also a prime.

Original entry on oeis.org

2, 37, 83, 137, 229, 317, 409, 557, 677, 829, 991, 1187, 1423, 1597, 1871, 2083, 2347, 2633, 2939, 3307, 3581, 3967, 4297, 4673, 5051, 5479, 5927, 6343, 6791, 7349, 7757, 8269, 8783, 9323, 9871, 10463, 11069, 11633, 12251, 12889, 13537, 14207, 14891, 15641

Offset: 1

Amiram Eldar, Aug 02 2019

Luhn primes were named after Norman Luhn, who first noted the property of 229 on the website Prime Curios!.

There are no Luhn primes in odd base, and only one, 2, in base 2.

			a(2) = 37 since 37 = 211 in base 2*2 = 4, and 211+112 = 323 which equals 59 in base 10 and is prime.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Octavian Cira and Florentin Smarandache, Luhn prime numbers, Theory and Applications of Mathematics & Computer Science, Vol. 5, No. 1 (2015), pp. 1-8.
G. L. Honaker, Jr. and Chris Caldwell, eds., 229, Prime Curios!, November 19, 2001.

Cf. A061783.

a[b_] := Module[{p=2}, While[!PrimeQ[p + FromDigits[Reverse @ IntegerDigits[p, b], b]], p = NextPrime[p]]; p]; Table[a[n], {n, 2, 88, 2}]

a(n) = {my(p=2); while (!isprime(p+fromdigits(Vecrev(digits(p, 2*n)), 2*n)), p = nextprime(p+1)); p;} \\ Michel Marcus, Aug 03 2019

a(n) > 8*n^2 for n > 1.

cp's OEIS Frontend

A086002 Primes which when added to their own rotation yield a prime.

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Maple

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Extensions

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

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Maple

Mathematica

PARI

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

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PARI

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A232446 Primes p such that reversal( p^2 ) + p is also prime.

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A304390 Prime numbers p such that p squared + (p reversed) squared is also prime.

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PARI

Python

A367793 Primes p such that the sum of p and its reversal is a semiprime.

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Maple

Mathematica

A174402 Primes such that applying "reverse and add" twice produces two more primes.

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