A004087 -id:A004087 - cp's OEIS frontend

A061783 Luhn primes: primes p such that p + (p reversed) is also 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, 20011, 20029, 20047, 20051, 20101, 20161, 20201, 20249, 20269, 20347, 20389, 20399, 20441, 20477, 20479, 20507

Offset: 1

Amarnath Murthy, May 24 2001

a(n) has an odd number of digits, as otherwise a(n) + reverse(a(n)) is a multiple of 11. For a(n) > 10, a(n) is prime and thus odd, and therefore the first digit of a(n) is even as otherwise a(n) + reverse(a(n)) is even and composite. - Chai Wah Wu, Aug 19 2015
See A072385 for the resulting primes p + reverse(p) = A056964(p). - M. F. Hasler, Sep 26 2019
Named by Cira and Smarandache (2014) after Norman Luhn, who noted the property of the prime 229 on the Prime Curios! website. - Amiram Eldar, Jun 05 2021

			229 is a term since 229 is a prime and so is 229 + 922 = 1151.

Harry J. Smith and Chai Wah Wu, Table of n, a(n) for n = 1..50598, giving all terms below 9*10^6 (The first 1000 terms from Harry J. Smith)
Octavian Cira and Florian Smarandache, Luhn prime numbers, Theory and Applications of Mathematics & Computer Science, Vol. 5, No. 1 (2015), pp. 29-36; preprint, 2014.
G. L. Honaker, Jr. and Chris Caldwell, eds., 229, Prime Curios!, November 19, 2001.
Chai Wah Wu, 3010506 terms, 11MB zipped file of all terms below 10^9.

Cf. A004086 (reverse), A004087 (primes reversed), A056964 (reverse & add), A072385, A086002 (similar, using "rotate" instead of "reverse").

[NthPrime(n): n in [1..2400] | IsPrime(s) where s is NthPrime(n)+Seqint(Reverse(Intseq(NthPrime(n))))]; // Bruno Berselli, Aug 05 2013

Select[Prime[Range[3000]],PrimeQ[#+FromDigits[Reverse[IntegerDigits[#]]]]&] (* Harvey P. Dale, Nov 27 2010 *)

isok(p) = { isprime(p) && isprime(p + fromdigits(Vecrev(digits(p)))) } \\ Harry J. Smith, Jul 28 2009

select( is_A061783(p)=isprime(A056964(p)) && isprime(p), primes(8713)) \\  A056964(p)=p+fromdigits(Vecrev(digits(p))). There is no term with 4 digits or starting with an odd digit, i.e., no candidate between prime(168) = 997 and prime(2263) = 20011. Using primes up to prime(8713) = 89989 ensures the list of 5-digit terms is complete. - M. F. Hasler, Sep 26 2019

from sympy import isprime, prime
A061783 = [prime(n) for n in range(1,10**5) if isprime(prime(n)+int(str(prime(n))[::-1]))] # Chai Wah Wu, Aug 14 2014

Corrected and extended by Patrick De Geest, May 26 2001

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

A071786 In prime factorization of n replace each prime with its reversal (in decimal notation).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 14, 15, 16, 71, 18, 91, 20, 21, 22, 32, 24, 25, 62, 27, 28, 92, 30, 13, 32, 33, 142, 35, 36, 73, 182, 93, 40, 14, 42, 34, 44, 45, 64, 74, 48, 49, 50, 213, 124, 35, 54, 55, 56, 273, 184, 95, 60, 16, 26, 63, 64, 155, 66, 76, 284, 96, 70, 17, 72

Offset: 1

Reinhard Zumkeller, Jun 06 2002

The range of A007500 is a subset of the range of this sequence. - Reinhard Zumkeller, Jul 06 2009

Prime factors counted with multiplicity. - Harvey P. Dale, Jul 08 2017

			a(143) = a(11*13) = a(11)*a(13) = 11*31 = 341.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

Cf. A151764, A161594, A151765. For records see A151766, A151767.
Cf. A151768 (complement), A376858 (fixed points).
Cf. A027746.

a071786 = product . map a004086 . a027746_row
-- Reinhard Zumkeller, Oct 14 2011

read("transforms") ; A071786 := proc(n) local ifs, a, d ; ifs := ifactors(n)[2] ; a := 1 ; for d in ifs do a := a*digrev(op(1, d))^op(2, d) ; od: a ; end: # R. J. Mathar, Jun 16 2009
# second Maple program:
r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):
a:= n-> mul(r(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 19 2017

Table[Times@@IntegerReverse/@Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]],{n,80}] (* Harvey P. Dale, Jul 08 2017 *)

rev(n)=fromdigits(Vecrev(digits(n)))
a(n)=my(f=factor(n)); prod(i=1,#f~,rev(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import factorint
from operator import mul
from functools import reduce
def A071786(n):
    return 1 if n==1 else reduce(mul,(int(str(p)[::-1])**e for p,e in factorint(n).items())) # Chai Wah Wu, Aug 14 2014

Completely multiplicative with a(p) = A004086(p), p prime.

a(A000040(n)) = A004087(n).

A133019 Product of n-th prime and n-th prime written backwards.

Original entry on oeis.org

4, 9, 25, 49, 121, 403, 1207, 1729, 736, 2668, 403, 2701, 574, 1462, 3478, 1855, 5605, 976, 5092, 1207, 2701, 7663, 3154, 8722, 7663, 10201, 31003, 75007, 98209, 35143, 91567, 17161, 100147, 129409, 140209, 22801, 117907, 58843, 127087

Offset: 1

Omar E. Pol, Oct 27 2007

a(8) = 1729 is the second taxicab number, also called the Hardy-Ramanujan number (see A001235, A011541 and A133029).

			a(8) = 1729 because the 8th prime is 19 and 19 written backwards is 91 and 19*91 = 1729.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A001235, A004087, A011541, A061205, A067563, A067087, A133029. Prime numbers: A000040.

#*FromDigits[Reverse[IntegerDigits[#]]] & /@ Prime[Range[1, 50]] (* G. C. Greubel, Oct 02 2017 *)
#*IntegerReverse[#]&/@Prime[Range[40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 29 2021 *)

vector(60, n, prime(n)*subst(Polrev(digits(prime(n))), x, 10)) \\ Michel Marcus, Dec 17 2014

a(n) = A000040(n) * A004087(n)

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

Original entry on oeis.org

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

A188649 Least common multiple of reversals of divisors of n in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 84, 31, 574, 255, 488, 71, 162, 91, 20, 84, 22, 32, 168, 260, 62, 72, 1148, 92, 510, 13, 11224, 33, 6106, 1855, 2268, 73, 15106, 93, 40, 14, 6888, 34, 44, 4590, 64, 74, 10248, 658, 260, 1065, 3100, 35, 3240, 55, 149240, 6825, 7820, 95, 7140, 16, 26, 252, 11224, 8680, 66, 76, 12212, 96, 152110, 17, 4536, 37, 6862, 251940, 2024204, 77

Offset: 1

Reinhard Zumkeller, Apr 11 2011

			Divisors(42) = {1,2,3,6,7,14,21,42}, therefore a(42) = lcm(1,2,3,6,7,41,12,24) = 6888.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A003990 (LCM), A027750 (divisors), A004086 (reversal), A188650.

Cf. A000040, A004087, A002385.

a188649 n = foldl lcm 1 $ map a004086 $ filter ((== 0) . mod n) [1..n]

a(n) = lcm(apply(x->fromdigits(Vecrev(digits(x))), divisors(n))); \\ Michel Marcus, Mar 13 2018

from math import lcm
from sympy import divisors
def a(n): return lcm(*(int(str(d)[::-1]) for d in divisors(n)))
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Aug 14 2022

a(A000040(n)) = A004087(n);

a(A002385(n)) = A002385(n), see A188650 for all fixed points.

A166501 Primes p such that (p reversed)+4 is also a prime.

Original entry on oeis.org

3, 7, 31, 73, 79, 97, 313, 331, 349, 379, 397, 541, 571, 709, 739, 757, 769, 937, 967, 3037, 3061, 3067, 3121, 3163, 3187, 3217, 3229, 3253, 3313, 3319, 3361, 3433, 3457, 3529, 3547, 3613, 3631, 3643, 3673, 3739, 3769, 3847, 3889, 5011, 5023, 5101, 5107

Offset: 1

Vincenzo Librandi, Oct 15 2009

			73 is in the sequence because 37+4=41 prime.
331 is in the sequence because 133+4=137 prime.
3457 is in the sequence because 7543+4=7547 prime.

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

[p: p in PrimesUpTo(6000) | IsPrime(q+4) where q is Seqint(Reverse(Intseq(p)))]; // Vincenzo Librandi, Sep 15 2013

Select[Prime[Range[700]],PrimeQ[FromDigits[Reverse[IntegerDigits[#]]]+ 4]&] (* Harvey P. Dale, May 21 2012 *)
Select[Prime[Range[700]],PrimeQ[IntegerReverse[#]+4]&] (* Harvey P. Dale, Mar 16 2023 *)

{p = A000040(i): A004087(i)+4 in A000040}. - R. J. Mathar, Oct 16 2009

Keyword:base added, 3163 inserted, sequence extended by R. J. Mathar, Oct 16 2009

A068396 n-th prime minus its reversal.

Original entry on oeis.org

0, 0, 0, 0, 0, -18, -54, -72, -9, -63, 18, -36, 27, 9, -27, 18, -36, 45, -9, 54, 36, -18, 45, -9, 18, 0, -198, -594, -792, -198, -594, 0, -594, -792, -792, 0, -594, -198, -594, -198, -792, 0, 0, -198, -594, -792, 99, -99, -495

Offset: 1

Reinhard Zumkeller, Mar 08 2002

a(n) = 0 for n in A075807. - Michel Marcus, Sep 27 2017

All terms are divisible by 9. - Zak Seidov, Jun 05 2021

			a(10) = 29 - 92 = -63;
a(20) = 71 - 17 = 54.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A004086, A056965, A061227, A075807.

a068396 n = p - a004086 p  where p = a000040 n
-- Reinhard Zumkeller, Feb 04 2014

#-IntegerReverse[#]& /@ Prime[Range[50]] (* Harvey P. Dale, Dec 20 2012 *)

a(n) = prime(n) - fromdigits(Vecrev(digits(prime(n)))); \\ Michel Marcus, Sep 27 2017

from sympy import prime
def a(n): pn = prime(n); return pn - int(str(pn)[::-1])
print([a(n) for n in range(1, 50)]) # Michael S. Branicky, Jun 05 2021

a(n) = A000040(n) - A004087(n).

a(n) = A056965(A000040(n)). - Michel Marcus, Sep 27 2017

A061227 a(n) = p + R(p) where R(p) is the digit reversal of n-th prime p.

Original entry on oeis.org

4, 6, 10, 14, 22, 44, 88, 110, 55, 121, 44, 110, 55, 77, 121, 88, 154, 77, 143, 88, 110, 176, 121, 187, 176, 202, 404, 808, 1010, 424, 848, 262, 868, 1070, 1090, 302, 908, 524, 928, 544, 1150, 362, 382, 584, 988, 1190, 323, 545, 949, 1151, 565, 1171, 383

Offset: 0

Amarnath Murthy, Apr 22 2001

			a(4) = 14 = 7 + 7, 7 is the fourth prime; a(8) = 110 = 19 + 91, 19 is the eighth prime.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000040, A004086, A056964.

a061227 n = p + a004086 p  where p = a000040 n
-- Reinhard Zumkeller, Feb 04 2014

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(10^(nops(L)-j)*L[j],j=1..nops(L))
end proc:
seq(x+revdigs(x),x=select(isprime,[2,seq(i,i=3..1000,2); # Robert Israel, May 23 2016

#+FromDigits[Reverse[IntegerDigits[#]]]&/@Prime[Range[60]] (* Harvey P. Dale, Jul 13 2013 *)

a(n) = A000040(n) + A004087(n). - Reinhard Zumkeller, Feb 04 2014

a(n) = A056964(A000040(n)). - Robert Israel, May 23 2016

More terms from Patrick De Geest, Jun 04 2001

A071602 Sum of the reverses of the first n primes.

Original entry on oeis.org

2, 5, 10, 17, 28, 59, 130, 221, 253, 345, 358, 431, 445, 479, 553, 588, 683, 699, 775, 792, 829, 926, 964, 1062, 1141, 1242, 1543, 2244, 3145, 3456, 4177, 4308, 5039, 5970, 6911, 7062, 7813, 8174, 8935, 9306, 10277, 10458, 10649, 11040, 11831

Offset: 1

Joseph L. Pe, Jun 02 2002

			a(6) = reverse(2) + reverse(3) + reverse(5) + reverse(7) + reverse(11) + reverse(13) = 2 + 3 + 5 + 7 + 11 + 31 = 59.

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 178. Shallit Minimal Primes Set, The Prime Puzzles & Problems Connection.
Jeffrey Shallit, Minimal Primes, Journal of Recreational Mathematics, vol. 30.2 1999-2000 pp. 113-117, Baywood NY.

Cf. A007504, A114835.

Partial sums of A004087.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
ListTools:-PartialSums(map(rev, [seq(ithprime(i),i=1..200)])); # Robert Israel, Feb 17 2025

f[n_] := Sum[ FromDigits[ Reverse[ IntegerDigits[Prime[i]]]], {i, 1, n}]; Table[ f[n], {n, 1, 50}]
Accumulate[FromDigits[Reverse[IntegerDigits[ #]]] & /@ Prime[Range[ 50]]]  (* Harvey P. Dale, Jan 27 2011 *)
Accumulate[IntegerReverse[Prime[Range[50]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 10 2020 *)

a(k) = my(v=primes(n)); sum(i=1, n, fromdigits(Vecrev(digits(v[i])))); \\ Michel Marcus, Feb 17 2025

from sympy import primerange
from itertools import accumulate
print(list(accumulate(int(str(p)[::-1]) for p in primerange(2, 198)))) # Michael S. Branicky, Jun 24 2022

Edited by Robert G. Wilson v, Jun 07 2002

A095180 Reverse digits of primes, append to sequence if result is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 13, 73, 17, 37, 97, 79, 101, 701, 311, 131, 941, 151, 751, 761, 971, 181, 191, 991, 113, 313, 733, 743, 353, 953, 373, 383, 983, 107, 907, 727, 337, 937, 347, 157, 757, 167, 967, 787, 797, 709, 919, 929, 739, 149, 359, 769, 179, 389, 199

Offset: 1

Cino Hilliard, Jun 21 2004

Conjecture: the Benford law limit is 2=Sum[N[Log[10, 1 + 1/d[[n]]]], {n, 1, Length[d]}]^2/(( #totalprimes/#totalPrimes)). At 50000 primes total it is 2.05931. - Roger L. Bagula and Gary W. Adamson, Jul 02 2008

Presumably this does not satisfy Benford's law. - N. J. A. Sloane, Feb 09 2017

			The prime 107 in reverse is 701 which is prime.

Indranil Ghosh, Table of n, a(n) for n = 1..25000 (terms 1..206 from Roger L. Bagula and Gary W. Adamson)
Eric Weisstein's World of Mathematics, Benford's Law.
Index entries for sequences related to Benford's law

Cf. A007500.

Cf. A004087, A010051.

a095180 n = a095180_list !! (n-1)
a095180_list =filter ((== 1) . a010051) a004087_list
-- Reinhard Zumkeller, Oct 14 2011

b = Flatten[Table[If[PrimeQ[Sum[IntegerDigits[Prime[n]][[i]]*10^(i - 1), {i, 1, Length[IntegerDigits[Prime[n]]]}]], Sum[IntegerDigits[Prime[n]][[i]]*10^(i - 1), {i, 1, Length[IntegerDigits[Prime[n]]]}], {}], {n, 1,1000}]] (* Roger L. Bagula and Gary W. Adamson, Jul 02 2008 *)
Select[FromDigits[Reverse[IntegerDigits[#]]]&/@Prime[Range[300]],PrimeQ] (* Harvey P. Dale, May 05 2015 *)

r(n) = forprime(x=1,n,y=eval(rev(x));if(isprime(y),print1(y","))) \ Get the reverse of the input string rev(str) = { local(tmp,j,s); tmp = Vec(Str(str)); s=""; forstep(j=length(tmp),1,-1, s=concat(s,tmp[j])); return(s) }

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A061783 Luhn primes: primes p such that p + (p reversed) is also a prime.

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A071786 In prime factorization of n replace each prime with its reversal (in decimal notation).

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A133019 Product of n-th prime and n-th prime written backwards.

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

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A188649 Least common multiple of reversals of divisors of n in decimal representation.

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A166501 Primes p such that (p reversed)+4 is also a prime.

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A068396 n-th prime minus its reversal.

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