A058993 -id:A058993 - cp's OEIS frontend

A057708 Numbers m such that 2^m reversed is prime.

1, 4, 5, 7, 10, 17, 24, 37, 45, 55, 70, 77, 107, 137, 150, 271, 364, 1157, 1656, 2004, 2126, 3033, 3489, 3645, 4336, 6597, 7279, 12690, 13840, 20108, 21693, 28888, 84155, 102930

Offset: 1

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

a(35) > 105000. - Giovanni Resta, Feb 22 2013
From Bernard Schott, Jan 30 2022: (Start)
If m is an even term, then u = m/2 is a term of A350441, this because 2^m = 4^(m/2). In fact, terms of A350441 are half the even terms of this sequence here.
If m is a term multiple of 3, then k = m/3 is a term of A350442, this because 2^m = 8^(m/3). First examples: m = 24, 45, 150, 1656, ... and corresponding k = 8, 15, 50, 552, ... (End)
a(35) > 200000. - Michael S. Branicky, May 12 2025

			4 is a term because 2^4 reversed is 61 and prime.

Cf. A004094, A341713.

Numbers m such that k^m reversed is prime: this sequence (k=2), A350441 (k=4), A058993 (k=5), A058994 (k=7), A350442 (k=8), A058995 (k=13).

with(numtheory): myarray := []: for n from 1 to 4000 do it1 := convert(2^n, base, 10): it2 := sum(10^(nops(it1)-i)*it1[i], i=1..nops(it1)): if isprime(it2) then printf(`%d,`,n) fi: od:

Do[ If[ PrimeQ[ FromDigits[ Reverse[ IntegerDigits[2^n]] ]], Print[ n]], {n, 20000}] (* Robert G. Wilson v, Jan 29 2005 *)
Select[Range[4400],PrimeQ[IntegerReverse[2^#]]&] (* Requires Mathematica version 10 or later *) (* The program generates the first 25 terms of the sequence; to generate more, increase the Range constant, but the program will take longer to run. *) (* Harvey P. Dale, Aug 05 2020 *)

isok(m) = isprime(fromdigits(Vecrev(digits(2^m)))) \\ Mohammed Yaseen, Jul 20 2022

from sympy import isprime
k, m, A057708_list = 1, 2,  []
while k <= 10**3:
    if isprime(int(str(m)[::-1])):
        A057708_list.append(k)
    k += 1
    m *= 2 # Chai Wah Wu, Mar 09 2021

More terms from Chris Nash (chris_nash(AT)prodigy.net), Oct 25 2000
Two more terms from Robert G. Wilson v, Jan 29 2001
3 more terms from Farideh Firoozbakht, Aug 05 2004
a(33)-a(34) from Giovanni Resta, Feb 22 2013

A058994 Numbers m such that 7^m reversed is prime.

Original entry on oeis.org

1, 12, 24, 225, 392, 819, 1201, 1645, 1775, 37578

Offset: 1

Robert G. Wilson v, Jan 17 2001

Cf. A059208, A071584.

Numbers m such that k^m reversed is prime: A057708 (k=2), A350441 (k=4), A058993 (k=5), this sequence (k=7), A350442 (k=8), A058995 (k=13).

Do[ If[ PrimeQ[ ToExpression[ StringReverse[ ToString[7^n] ] ] ], Print[n] ], {n, 1, 2500} ]

isok(m) = isprime(fromdigits(Vecrev(digits(7^m)))) \\ Mohammed Yaseen, Jul 20 2022

from sympy import isprime
k, m, A058994_list = 1, 7,  []
while k <= 10**3:
    if isprime(int(str(m)[::-1])):
        A058994_list.append(k)
    k += 1
    m *= 7 # Chai Wah Wu, Mar 09 2021

a(10) from Michael S. Branicky, Mar 16 2025

A058995 Numbers m such that 13^m reversed is prime.

Original entry on oeis.org

1, 379, 467, 479, 1325, 7144, 10311, 26079

Offset: 1

Robert G. Wilson v, Jan 17 2001

Numbers m such that k^m reversed is prime: A057708 (k=2), A350441 (k=4), A058993 (k=5), A058994 (k=7), A350442 (k=8), this sequence (k=13).

Do[ If[ PrimeQ[ ToExpression[ StringReverse[ ToString[13^n] ] ] ], Print[n] ], {n, 1, 2300} ]
Select[Range[1400],PrimeQ[IntegerReverse[13^#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 18 2020 *)

isok(m) = isprime(fromdigits(Vecrev(digits(13^m)))) \\ Mohammed Yaseen, Jul 21 2022

from sympy import isprime
k, m, A058995_list = 1, 13,  []
while k <= 10**3:
    if isprime(int(str(m)[::-1])):
        A058995_list.append(k)
    k += 1
    m *= 13 # Chai Wah Wu, Mar 09 2021

a(6)-a(7) from Chai Wah Wu, Mar 09 2021

a(8) from Michael S. Branicky, Apr 01 2023

A085298 a(n) is the smallest exponent x such that prime(n)^x when reversed is a prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 8, 7, 1, 1, 2, 5, 15, 10, 12, 4, 39, 1, 1, 1, 11, 2, 1, 1, 10, 1, 23, 1, 5, 1, 243, 2, 1, 1, 1, 23, 1, 34, 1, 1, 1, 2, 58, 1, 3, 9, 166, 17, 68, 8, 8, 3, 7, 5, 5, 2, 2, 2, 61, 11, 97, 1, 1, 10, 2, 1, 1, 41, 1, 1, 66, 1, 5, 1, 1, 2, 2, 8, 40, 2, 8, 19, 2, 2, 723

Offset: 1

Labos Elemer, Jun 24 2003

It is conjectured that for every n such exponent exists.

			a(n)=1 means that rev(prime(n)) is prime i.e. prime(n) is in A007500;
a(n)=2 means that rev(prime(n)^2) is prime but rev(prime(n)) is not, like n=8:p=19 and 91 is not a prime but rev[19^2]=rev[361]=163 is a prime;
For n, the first k exponent providing rev(prime(n)^k) prime can be quite large, like at n=87: rev(p(87)^723)=rev(449^723) is the first [probably] prime has 1918 decimal digits: 948......573.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A000040, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

a:= proc(n) local k, p; p:= ithprime(n); for k while not isprime((s->
      parse(cat(seq(s[-i], i=1..length(s)))))(""||(p^k))) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 04 2019

a[n_] := Block[{k = 1}, While[! PrimeQ@ FromDigits@ Reverse@ IntegerDigits[ Prime[n]^k], k++]; k]; Array[a, 87] (* Giovanni Resta, Sep 04 2019 *)

a(n) = {my(x=1, p=prime(n)); while (!ispseudoprime(fromdigits(Vecrev(digits(p^x)))), x++); x;} \\ Michel Marcus, Sep 04 2019

a(n) = Min{x; reversed(prime(n)^x) is a prime}.

A350441 Numbers m such that 4^m reversed is prime.

Original entry on oeis.org

2, 5, 12, 35, 75, 182, 828, 1002, 1063, 2168, 6345, 6920, 10054, 14444, 51465

Offset: 1

Mohammed Yaseen, Dec 31 2021

From Bernard Schott, Jan 30 2022: (Start)
If m is a term, then u = 2*m is a term of A057708, because 4^m = 2^(2*m). In fact, terms of this sequence here are half the even terms of A057708.
If m is a term that is multiple of 3, then k = 2*m/3 is a term of A350442, because 4^m = 8^(2m/3). First examples: m = 12, 75, 828, 1002, 6345, 51465, ... and corresponding k = 8, 50, 552, 668, 4230, 34310, ... (End)

Cf. A058996, A071582.

Cf. Numbers m such that k^m reversed is prime: A057708 (k=2), this sequence (k=4), A058993 (k=5), A058994 (k=7), A350442 (k=8), A058995 (k=13).

Select[Range[2200], PrimeQ[IntegerReverse[4^#]] &] (* Amiram Eldar, Dec 31 2021 *)

isok(m) = isprime(fromdigits(Vecrev(digits(4^m))))

from sympy import isprime
m = 4
for n in range (1, 2000):
    if isprime(int(str(m)[::-1])):
        print(n)
    m *= 4

a(11)-a(15) from Amiram Eldar, Dec 31 2021

A350442 Numbers m such that 8^m reversed is prime.

Original entry on oeis.org

8, 15, 50, 552, 668, 1011, 1163, 1215, 2199, 4230, 7231, 34310

Offset: 1

Mohammed Yaseen, Dec 31 2021

From Bernard Schott, Jan 30 2022: (Start)
If k is a term, then u = 3*k is a term of A057708, because 8^k = 2^(3k).
If k is an even term, then t = 3*k/2 is a term of A350441, because 8^k = 4^(3k/2). First examples: k = 8, 50, 552, 668, 4230, 34310, ... and corresponding t = 12, 75, 828, 1002, 6345, 51465, ... (End)

Cf. A059003, A071586.

Cf. Numbers m such that k^m reversed is prime: A057708 (k=2), A350441 (k=4), A058993 (k=5), A058994 (k=7), A058995 (k=13).

Select[Range[2200], PrimeQ[IntegerReverse[8^#]] &] (* Amiram Eldar, Dec 31 2021 *)

isok(m) = isprime(fromdigits(Vecrev(digits(8^m))))

from sympy import isprime
m = 8
for n in range (1, 2000):
    if isprime(int(str(m)[::-1])):
        print(n)
    m *= 8

a(10)-a(12) from Amiram Eldar, Dec 31 2021

A085300 a(n) is the least prime x such that when reversed it is a power of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 163, 18258901387, 90367894271, 13, 73, 1861, 344800741, 34351783286302805384336021, 940315563074788471, 1886172359328147919771, 14854831

Offset: 1

Labos Elemer, Jun 24 2003

A006567 (after rearranging terms) and A002385 are subsequences. - Chai Wah Wu, Jun 02 2016

			a(14)=344800741 means that 147008443=43^5=p(14)^5, where 5 is the smallest such exponent;
a(19) has 82 decimal digits and if reversed equals 39th power of p(19)=67.

Chai Wah Wu, Table of n, a(n) for n = 1..86

Cf. A085298, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

from sympy import prime, isprime
def A085300(n):
    p = prime(n)
    q = p
    while True:
        m = int(str(q)[::-1])
        if isprime(m):
            return(m)
        q *= p # Chai Wah Wu, Jun 02 2016

A085299 a(n) is the smallest number x such that A085298[x]=n, or 0 if no such number exists.

Original entry on oeis.org

1, 8, 47, 18, 14, 89, 10, 9, 48, 16, 23, 17, 168, 268, 15, 661, 50, 380, 84, 116, 360, 245, 29, 144, 345, 227, 785, 261, 148, 235, 691, 658, 638, 40, 1023, 674, 1529, 210, 19, 81, 181, 428, 170, 1130, 2322, 406, 600, 373, 958, 217

Offset: 1

Labos Elemer, Jun 24 2003

			a(13) = 168 means that 13 is the smallest exponent such that reversed[p(168)^13] = reversed[997^13] = 776831144302925059735912605306533496169
is prime if read in this direction and 13th prime-power if read backwards.

Cf. A085298, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

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A057708 Numbers m such that 2^m reversed is prime.

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A058994 Numbers m such that 7^m reversed is prime.

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A058995 Numbers m such that 13^m reversed is prime.

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A085298 a(n) is the smallest exponent x such that prime(n)^x when reversed is a prime.

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A350441 Numbers m such that 4^m reversed is prime.

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A350442 Numbers m such that 8^m reversed is prime.

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A085300 a(n) is the least prime x such that when reversed it is a power of prime(n).

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A085299 a(n) is the smallest number x such that A085298[x]=n, or 0 if no such number exists.