A095179 -id:A095179 - cp's OEIS frontend

A371654 Numbers that are not multiples of 10 and that yield a prime if their digits are arranged in ascending order.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 37, 47, 59, 67, 71, 73, 74, 76, 79, 89, 91, 92, 95, 97, 98, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 172, 173, 175, 176, 179, 193, 194, 197, 199, 203, 209, 217, 223, 227, 229, 232, 233, 239, 257

Offset: 1

César Eliud Lozada, Apr 01 2024

If N is a term then all numbers with the same digits as N are terms too.

			91 is a term because arranging its digits in ascending order yields 19, which is a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007500, A095179, A371653.

Select[Range[500],  Mod[#, 10] != 0 && PrimeQ[FromDigits[Sort[IntegerDigits[#]]]] &]

from sympy import isprime
def ok(n): return n%10 and isprime(int("".join(sorted(str(n)))))
print([k for k in range(260) if ok(k)]) # Michael S. Branicky, Apr 01 2024

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A371654_gen(): # generator of terms
    for l in count(1):
        xlist = []
        for p in combinations_with_replacement('0123456789',l):
            if isprime(int(''.join(p))):
                xlist.extend(int(''.join(d)) for d in multiset_permutations(p) if d[0] != '0' and d[-1] != '0')
        yield from sorted(xlist)
A371654_list = list(islice(A371654_gen(),20)) # Chai Wah Wu, Apr 10 2024

A088112 Perfect powers whose digit reversal is prime.

Original entry on oeis.org

16, 32, 125, 128, 196, 361, 784, 1024, 1369, 1444, 1600, 1681, 3844, 7225, 7921, 9025, 9409, 11236, 14161, 18496, 19321, 19600, 36100, 37249, 38416, 70756, 73441, 75076, 76729, 78400, 78961, 97969, 99856, 102400, 105625, 107584, 109561, 111556

Offset: 1

Amarnath Murthy, Sep 25 2003

Conjectures: (1) Sequence is infinite. (2) For every n there are infinitely many members of the form k^n.

Harvey P. Dale, Table of n, a(n) for n = 1..471 (no further terms through 32 million)

Intersection of A001597 and A095179.

Select[Range[120000],GCD@@FactorInteger[#][[All,2]]>1 && PrimeQ[ IntegerReverse[ #]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 03 2018 *)

Corrected and extended by Ray Chandler, Oct 14 2003

Offset changed by Andrew Howroyd, Sep 22 2024

A253912 Fourth powers whose reversal is a prime.

Original entry on oeis.org

16, 38416, 130321, 160000, 923521, 1500625, 13845841, 14776336, 16777216, 38950081, 163047361, 181063936, 312900721, 322417936, 384160000, 937890625, 1303210000, 1600000000, 3722098081, 7992538801, 9235210000, 13841287201, 15006250000, 16610312161, 17748900625, 31414372081, 37141383841

Offset: 1

Rabii Younès, Jan 18 2015

Intersection of A000583 and A095179.
As the last digits of primes are not even or 5 (except for primes 2 and 5), the terms do not start with an even number or 5. If m is an integer such that the reversal of m^4 is prime and sqrt4(n) is the fourth root of n then m is not of the form [sqrt4(2 * 10^k), sqrt4(3 * 10^k)] or [sqrt4(4 * 10^k), sqrt4(7 * 10^k)] for nonnegative k etc. - David A. Corneth, Jun 02 2016
All terms == 1 mod 3. - Robert Israel, Jun 03 2016

			Example: a(1) = 16 is a fourth power because 16 = 2^4 and the reverse of 16 is 61 which is a prime number.

Chai Wah Wu, Table of n, a(n) for n = 1..25659

Cf. A000583 (fourth power), A095179 (reversal of n is prime).

Cf. A058996 (primes whose reversal is a fourth power).

Select[Range[440]^4, PrimeQ[FromDigits@ Reverse@ IntegerDigits@ #] &] (* Michael De Vlieger, Jan 19 2015 *)

from sympy import isprime
A253912_list = [n for n in (i**4 for i in range(10**6)) if isprime(int(str(n)[::-1]))] # Chai Wah Wu, Jun 02 2016

A257636 Numbers n such that the base 10 reversals of n and n+1 are both prime.

Original entry on oeis.org

2, 13, 16, 30, 31, 34, 37, 70, 73, 91, 97, 106, 112, 118, 124, 130, 133, 145, 151, 166, 181, 199, 300, 310, 346, 358, 361, 364, 370, 376, 382, 388, 391, 700, 709, 721, 727, 730, 739, 745, 751, 754, 757, 760, 763, 775, 778, 784, 787, 790, 904, 907, 916, 919

Offset: 1

Robert Israel, Nov 04 2015

n such that n and n+1 are in A095179.
Leading 0's in the reversals are allowed.
Heuristically, the abundance of these numbers should be roughly similar to that of the twin primes.  Thus the sequence should be infinite but the sum of the reciprocals should converge.
All terms == 1 (mod 3) except for 2 and 3*10^k where k is in A049054.

			13 is in the sequence because both 31 and 41 are prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Math StackExchange, Consecutive numbers where their revers numbers are primes

Cf. A000040, A004086, A049054, A095179.

revdigs:= proc(n) option remember; local x;
   x:= n mod 10;
   x*10^ilog10(n)+revdigs((n-x)/10);
end proc:
for i from 0 to 9 do revdigs(i):= i od:
Rprimes:= select(isprime@revdigs, [$1..10^4]):
Rprimes[select(t -> Rprimes[t+1]-Rprimes[t]=1, [$1..nops(Rprimes)-1])]; # Robert Israel, Nov 04 2015

SequencePosition[Table[If[PrimeQ[IntegerReverse[n]],1,0],{n,1000}],{1,1}][[;;,1]] (* Harvey P. Dale, Jan 07 2024 *)

for(n=1, 1e3, if(isprime(eval(concat(Vecrev(Str(n))))) && isprime(eval(concat(Vecrev(Str(n+1))))), print1(n, ", "))) \\ Altug Alkan, Nov 04 2015

A354881 Primes p such that, if q is the next prime, the digit reversal of p*q is prime.

Original entry on oeis.org

5, 31, 37, 41, 53, 103, 197, 263, 277, 337, 349, 353, 359, 373, 397, 401, 421, 431, 439, 547, 569, 587, 599, 857, 859, 863, 877, 883, 983, 1009, 1013, 1039, 1069, 1091, 1097, 1103, 1117, 1129, 1153, 1171, 1193, 1213, 1223, 1237, 1249, 1279, 1291, 1301, 1367, 1811, 1871, 1931, 1979, 2647, 2663

Offset: 1

J. M. Bergot and Robert Israel, Jul 13 2022

Primes p such that A013636(p) is in A095179.

			a(3) = 37 is a term because 37 is prime, the next prime is 41, 37*41 = 1517 and its digit reversal 7151 is prime.

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

Cf. A013636, A004086, A095179.

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
P:= [seq(ithprime(i),i=1..1000)]:
P[select(i -> isprime(revdigs(P[i]*P[i+1])), [$1..999])];

a354881[n_] := Select[Map[Prime, Range[n]], PrimeQ[FromDigits[Reverse[IntegerDigits[# NextPrime[#]]]]]&]
a354881[390] (* Hartmut F. W. Hoft, Jul 20 2022 *)
Select[Partition[Prime[Range[400]],2,1],PrimeQ[IntegerReverse[Times@@#]]&][[;;,1]] (* Harvey P. Dale, Feb 10 2024 *)

A272022 Look at the set of numbers obtained by permuting the digits of n in all possible ways, then remove n itself from the set. If the remaining numbers are all primes, then n is in the sequence.

Original entry on oeis.org

13, 14, 16, 17, 20, 30, 31, 32, 34, 35, 37, 38, 50, 70, 71, 73, 74, 76, 79, 91, 92, 95, 97, 98, 110, 113, 118, 119, 131, 133, 199, 311, 337, 373, 733, 772, 775, 778, 779, 919, 991, 1118, 3337, 7771, 77779

Offset: 1

César Eliud Lozada, Apr 18 2016

If it exists, a(46) > 5*10^11. - Lars Blomberg, Mar 31 2018

			119 is in the sequence because every permutation of its digits excluding 119 (i.e., 191 and 911) is a prime.
11 is not in the sequence, because when 11 is removed from the set, no numbers are left.

Cf. A095179, A111347.

Cf. A003459. - Altug Alkan, Apr 18 2016

lis := [];
for n from 1 to 10000 do
  nn := convert(n, base, 10);
  pp := combinat[permute](nn);
  if nops(pp) = 1 then
    next
  end if;
  lOk := true;
  for p in pp do
    if p = nn then
      next: #exclude n
    end if;
    if `not`(isprime(convert(p, base, 10, 10^nops(p))[])) then
      lOk := false; break
    end if
  end do;
  if lOk then
    lis := [op(lis), n]
  end if
end do:
lis := lis;

rnapQ[n_]:=Module[{p=Rest[FromDigits/@Permutations[IntegerDigits[ n]]]},If[ Length[p]==0, False, AllTrue[p,PrimeQ]]]; Select[Range[80000],rnapQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 24 2019 *)

isok(n) = {v = []; d = digits(n); for (k=0, (#d)!-1, p = numtoperm(#d, k); dp = vector(#d, j, d[p[j]]); np = subst(Pol(dp), x, 10); v = Set(concat(v, np));); v = setminus(v, Set(n)); if (#v == 0, return (0)); for (k=1, #v, if (!isprime(v[k]), return (0));); return (1);} \\ Michel Marcus, Apr 18 2016

from sympy import isprime
from itertools import count, islice, permutations
def agen(): yield from (k for k in count(1) if len(set(s:=str(k)))!=1 and all((t:=int("".join(m)))==k or isprime(t) for m in permutations(s)))
print(list(islice(agen(), 45))) # Michael S. Branicky, Dec 29 2023

cp's OEIS Frontend

A371654 Numbers that are not multiples of 10 and that yield a prime if their digits are arranged in ascending order.

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A088112 Perfect powers whose digit reversal is prime.

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A253912 Fourth powers whose reversal is a prime.

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A257636 Numbers n such that the base 10 reversals of n and n+1 are both prime.

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A354881 Primes p such that, if q is the next prime, the digit reversal of p*q is prime.

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A272022 Look at the set of numbers obtained by permuting the digits of n in all possible ways, then remove n itself from the set. If the remaining numbers are all primes, then n is in the sequence.

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Mathematica

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Python