A007488 -id:A007488 - cp's OEIS frontend

A132388 Same as A007488, but with the numbers arranged so that their reversals are in increasing order.

61, 691, 163, 487, 4201, 9631, 4441, 1861, 4483, 5227, 1297, 5209, 9049, 63211, 16141, 69481, 12391, 94273, 61483, 65707, 14437, 67057, 92767, 16987, 96979, 65899, 526501, 485701, 165901, 655111, 522211, 906421, 520621, 123031, 440131, 921931, 921241

Offset: 1

N. J. A. Sloane, based on an email from Jeremy Gardiner, Nov 12 2007

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
NCES, Mind Benders [Source for sequence]

Cf. A068989.

read transforms: A132388ind := proc(n) option remember: local d,k: if(n=1)then return 4: fi: for k from procname(n-1)+1 do d:=digrev(k^2): if(length(k^2)=length(d) and isprime(digrev(k^2)))then return k: fi: od: end: A132388 := proc(n) return digrev(A132388ind(n)^2): end: seq(A132388(n),n=1..50); # Nathaniel Johnston, May 06 2011

Better description from Zak Seidov, Nov 14 2007

A059007 Numbers m such that m^2 reversed is a prime.

Original entry on oeis.org

4, 14, 19, 28, 32, 37, 38, 40, 41, 62, 85, 89, 95, 97, 106, 119, 136, 139, 140, 190, 193, 196, 266, 271, 274, 277, 280, 281, 313, 316, 320, 325, 328, 331, 334, 335, 353, 355, 361, 362, 370, 373, 377, 380, 383, 397, 398, 400, 401, 403, 410, 412, 421, 434, 439

Offset: 1

Robert G. Wilson v, Jan 16 2001

			28 is in the sequence because the reverse of 28^2 is 487 which is a prime. - _Indranil Ghosh_, Feb 10 2017

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (terms 0..1000 from T. D. Noe)

Cf. A007488.

Numbers m such that m^k reversed is a prime: A059008 (k=3), A059205 (k=4), A059206 (k=5), A059207 (k=6), A059208 (k=7), A059209 (k=8), A059210 (k=9), A059211 (k=10), A059212 (k=11), A059213 (k=12).

[n: n in [1..500] | IsPrime(Seqint(Reverse(Intseq(n^2))))]; // Marius A. Burtea, Jan 12 2019

Select[ Range[ 1000 ], PrimeQ[ ToExpression[ StringReverse[ ToString[ #^2 ] ] ] ] & ]
Select[Range[500],PrimeQ[IntegerReverse[#^2]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)

isok(n) = isprime(fromdigits(Vecrev(digits(n^2)))); \\ Michel Marcus, Jan 12 2019

A033294 Squares which when written backwards remain square (final 0's excluded).

Original entry on oeis.org

1, 4, 9, 121, 144, 169, 441, 484, 676, 961, 1089, 9801, 10201, 10404, 10609, 12321, 12544, 12769, 14641, 14884, 40401, 40804, 44521, 44944, 48841, 69696, 90601, 94249, 96721, 698896, 1002001, 1004004, 1006009, 1022121, 1024144, 1026169

Offset: 1

Jeff Burch

Of this sequence's first 10000 terms, only nine have an even number of digits; see A354256.

			144 = 12 * 12 is a term because 441 = 21 * 21.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
Index entry for sequences related to reversing digits of squares

Subsequence of A115690.

Cf. A007488, A007500, A061457, A354256.

a033294 n = a033294_list !! (n-1)
a033294_list = filter chi a000290_list where
  chi m = m `mod` 10 > 0 && head ds `elem` [1,4,5,6,9] &&
          a010052 (foldl (\v d -> 10 * v + d) 0 ds) == 1 where
    ds = unfoldr
         (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 10) m
-- Reinhard Zumkeller, Jan 19 2012

Select[Range[1100]^2,Mod[#,10]!=0&&IntegerQ[Sqrt[FromDigits[Reverse[ IntegerDigits[ #]]]]]&] (* Harvey P. Dale, Oct 28 2013 *)

from math import isqrt
from itertools import count, islice
def sqr(n): return isqrt(n)**2 == n
def agen():
    yield from (k*k for k in count(1) if k%10 and sqr(int(str(k*k)[::-1])))
print(list(islice(agen(), 36))) # Michael S. Branicky, May 21 2022

More terms from Erich Friedman

Initial 0 removed and offset changed by Reinhard Zumkeller, Jan 19 2012

A057699 Primes whose reversal is a cube.

Original entry on oeis.org

521, 806041, 969407, 1393939, 2303461, 3989683, 4831037, 5783273, 8081153, 8485181, 11520991, 15231851, 23206301, 25578253, 29925251, 32296051, 48762541, 52182343, 57369149, 61277761, 67134511, 67954643, 74825299

Offset: 1

G. L. Honaker, Jr., Oct 23 2000

Zak Seidov and Chai Wah Wu, Table of n, a(n) for n = 1..28987 n = 1..1500 from Zak Seidov.

Cf. A000040, A007488, A272692 (first differences).

Do[ If[ PrimeQ[ n ] && IntegerQ[ ToExpression[ StringReverse[ ToString[ n ] ] ]^(1/3) ], Print[ n ] ], {n, 1, 10^9} ]

flip(n)=fromdigits(Vecrev(digits(n)))
Set(select(isprime, vector(1000,n,flip(n^3)))) \\ Charles R Greathouse IV, Jun 07 2016

from sympy import isprime
A057699_list = []
for i in range(10**6):
    if i % 10:
        p = int(str(i**3)[::-1])
        if isprime(p):
            A057699_list.append(p)
A057699_list = sorted(A057699_list) # Chai Wah Wu, Jun 02 2016

A058996 Primes whose reversal is a fourth power.

Original entry on oeis.org

61, 61483, 123031, 125329, 5260051, 14854831, 18005983, 61277761, 63367741, 127009213, 163740361, 526098739, 639360181, 639714223, 1088352997, 1808902273, 10007631583, 10278214831, 14838314173, 16121301661

Offset: 1

Robert G. Wilson v, Jan 16 2001

Chai Wah Wu, Table of n, a(n) for n = 1..22582

Cf. A007488.

Do[ If[ PrimeQ[ n ] && IntegerQ[ ToExpression[ StringReverse[ ToString[ n ] ] ]^(1/4) ], Print[ n ] ], {n, 1, 10^12} ]

from sympy import isprime
A058996_list = []
for i in range(10**6):
    if i % 10:
        p = int(str(i**4)[::-1])
        if isprime(p):
            A058996_list.append(p)
A058996_list = sorted(A058996_list) # Chai Wah Wu, Jun 02 2016

A059008 Numbers n such that n^3 reversed is a prime.

Original entry on oeis.org

5, 50, 52, 89, 118, 122, 152, 155, 157, 194, 211, 218, 226, 244, 247, 248, 251, 256, 271, 325, 326, 328, 413, 452, 455, 463, 466, 467, 481, 485, 487, 491, 499, 500, 503, 520, 521, 523, 526, 541, 544, 547, 563, 571, 581, 584, 685, 686, 701, 707, 716, 721

Offset: 1

Robert G. Wilson v, Jan 16 2001

			52 is in the sequence because the reverse of 52^3 is 806041 and it is a prime. - _Indranil Ghosh_, Feb 10 2017

Indranil Ghosh, Table of n, a(n) for n = 1..1023

Cf. A007488.

[n: n in [1..800] | IsPrime(Seqint(Reverse(Intseq(n^3))))]; // Marius A. Burtea, Jan 12 2019

Select[ Range[ 1000 ], PrimeQ[ ToExpression[ StringReverse[ ToString[ #^3 ] ] ] ] & ]
Select[Range[1000],PrimeQ[IntegerReverse[#^3]]&] (* Harvey P. Dale, Dec 20 2023 *)

isok(n) = isprime(fromdigits(Vecrev(digits(n^3)))); \\ Michel Marcus, Jan 12 2019

A059003 Primes whose reversal is an eighth power.

Original entry on oeis.org

61277761, 18258901387, 526098218446813, 52609352682209503, 186386112766353931, 526093005165061333, 12785316583844897311, 18866248772202954601, 104414665083132185191, 125722028990735440387, 679091464527322606741, 1462701398540433470911, 6100159128959496276541

Offset: 1

Robert G. Wilson v, Jan 16 2001

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

Cf. A007488.

rev:= proc(n) local L, i;
  L:= convert(n, base, 10);
  add(L[-i]*10^(i-1), i=1..nops(L))
end proc:
S:= {}:
for i from 1 to 999 do
  if i mod 10 = 0 then next fi;
  p:= rev(i^8);
  if isprime(p) then S:= S union {p} fi;
od:
sort(convert(S, list)); # Robert Israel, Dec 22 2024

Do[ If[ PrimeQ[ n ] && IntegerQ[ ToExpression[ StringReverse[ ToString[ n ] ] ]^(1/8) ], Print[ n ] ], {n, 1, 10^22} ]

More terms from Sean A. Irvine, Sep 09 2022

A176371 Primes p such that reversal(p) - 13 is a square.

Original entry on oeis.org

31, 41, 71, 83, 281, 311, 431, 479, 733, 751, 797, 2011, 2857, 3163, 4373, 4397, 4943, 7541, 7577, 7583, 9413, 9491, 20533, 20731, 20771, 24151, 24547, 24767, 26249, 28979, 31121, 41201, 41609, 43321, 43391, 43753, 45641, 49459, 49463, 49811, 49891

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 16 2010

R(n) denotes the Reversal of a natural number n
List of all (p,N) for p < 10^6 - 1:
(*) for emirp pair (p,R(p)), (+) if square base N is a prime
(41,1), (71,2) (+) (*), (83,5) (+), (281,13) (+), (311,10) (*), (431,11) (+), (479,31) (+), (733,18) (*), (751,12) (*), (797,28),
(2011,33), (2857,87), (4373,61) (+), (4397,89) (+), (4943,59) (+), (7541,38), (7577,88) (*), (7583,62), (9413,56), (9491,44) (*), (20533,183), (20731,117), (20771,133), (24151,123), (24547,273), (24767,277) (+), (26249,307) (+), (28979,313) (+), (31121,110) (*), (41201,101) (+),
(41609,301), (43321,111), (43391,139) (+), (43753,189), (45641,121), (49459,309), (49463,191) (+), (49811,109), (49891,141), (71293,198) (*),
(73133,182), (73471,132), (73597,282) (*), (75521,112), (77611,108) (*), (77849,308), (77863,192) (*), (79613,178), (79841,122) (*), (83207,265),
(83231,115), (83243,185), (83299,315), (90031,114) (*), (92801,104), (96431,116) (*), (98057,274)

			41 = prime(13), R(41) - 13 = 14 - 13 = 1^2, is a term.
71 = prime(20), 17 - 13 = 2^2, is a term.
83 = prime(23), 38 - 13 = 5^2, is a term.
797 = prime(139) = palindromic prime(18), N = 28^2, is also a term.
Note successive terms that are also consecutive primes: p(17) = 7577, p(18) = 7583, p(36) = 49459, p(37) = 49463, p(46) = 77849, p(47) = 77863.

W. W. R. Ball, H. S. M.Coxeter: Mathematical Recreations and Essays, Dover Publications, 13th edition, 1987
O. Fritsche, R. Mischak and T. Krome: Verflixt und zugeknobelt, Mehr mathematische Raetselgeschichten, Rowohlt TB. Nr.62190, 2007
C. W. Trigg, Primes with Reverses That Are Powers, J. Rec. Math. 17, 1985

Chai Wah Wu, Table of n, a(n) for n = 1..7605

Cf. A000040, A000290, A002385, A006567, A007488.

isok(n) = {if (! isprime(n), return (0)); d = digits(n); revn = sum(i=1, #d, d[i]*10^(i - 1)); issquare(revn-13);} \\ Michel Marcus, Aug 25 2013

from sympy import isprime
A176371_list, i, j = [], 0, 13
while j < 10**10:
    p = int(str(j)[::-1])
    if j % 10 and isprime(p):
        A176371_list.append(p)
    j += 2*i+1
    i += 1
A176371_list = sorted(A176371_list) # Chai Wah Wu, Dec 17 2015

Two more terms 31 and 3163 added by Michel Marcus, Aug 25 2013

A059000 Primes whose reversal is a fifth power.

Original entry on oeis.org

23, 4201, 102658511, 344800741, 39715342481, 70496383033, 869910021839, 998699567381, 3457914828521, 3487946075153, 5265190686031, 5786421085169, 7020715917491, 9432574158041, 9925883645611, 9987727089187, 23802566907811, 23888027348153, 34401855516071

Offset: 1

Robert G. Wilson v, Jan 16 2001

Chai Wah Wu, Table of n, a(n) for n = 1..18022

Cf. A007488.

Do[ If[ PrimeQ[ n ] && IntegerQ[ ToExpression[ StringReverse[ ToString[ n ] ] ]^(1/5) ], Print[ n ] ], {n, 1, 10^16} ]

from sympy import isprime
A059000_list = []
for i in range(10**6):
    if i % 10:
        p = int(str(i**5)[::-1])
        if isprime(p):
            A059000_list.append(p)
A059000_list = sorted(A059000_list) # Chai Wah Wu, Dec 20 2015, Jun 02 2016

More terms from Sean A. Irvine, Sep 09 2022

A059001 Primes whose reversal is a sixth power.

Original entry on oeis.org

61277761, 10278214831, 424176600403, 526098190537, 526515941773, 5260934114929, 9481530370051, 40512620860813, 46882459723321, 108153140207347, 489332144054323, 526046241813643, 1619463705594643, 1676989428458959, 4249139677419331, 4878387447701941

Offset: 1

Robert G. Wilson v, Jan 16 2001

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

Cf. A007488.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
S:= {}:
for i from 1 to 999 do
  if i mod 10 = 0 then next fi;
  p:= rev(i^6);
  if isprime(p) then S:= S union {p}; fi;
od:
sort(convert(S,list)); # Robert Israel, Dec 22 2024

Do[ If[ PrimeQ[ n ] && IntegerQ[ ToExpression[ StringReverse[ ToString[ n ] ] ]^(1/6) ], Print[ n ] ], {n, 1, 10^18} ]

More terms from Sean A. Irvine, Sep 09 2022

cp's OEIS Frontend

A132388 Same as A007488, but with the numbers arranged so that their reversals are in increasing order.

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A059007 Numbers m such that m^2 reversed is a prime.

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Magma

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A033294 Squares which when written backwards remain square (final 0's excluded).

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Haskell

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A057699 Primes whose reversal is a cube.

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Mathematica

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A058996 Primes whose reversal is a fourth power.

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A059008 Numbers n such that n^3 reversed is a prime.

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Magma

Mathematica

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A059003 Primes whose reversal is an eighth power.

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Maple

Mathematica

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A176371 Primes p such that reversal(p) - 13 is a square.

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