A003459 -id:A003459 - cp's OEIS frontend

A354745 Non-repdigit numbers k such that every permutation of the digits of k has the same number of divisors.

13, 15, 17, 24, 26, 31, 37, 39, 42, 51, 58, 62, 71, 73, 79, 85, 93, 97, 113, 117, 131, 155, 171, 177, 178, 187, 199, 226, 262, 288, 311, 337, 339, 355, 373, 393, 515, 535, 551, 553, 558, 585, 622, 711, 717, 718, 733, 771, 781, 817, 828, 855, 871, 882, 899, 919, 933, 989, 991, 998

Offset: 1

Metin Sariyar, Jun 05 2022

After a(93) = 84444, no further terms < 10^18. - Michael S. Branicky, Jun 08 2022

			871 is a term because d(871) = d(817) = d(178) = d(187) = d(718) = d(781) = 4, where d(n) is the number of divisors of n.

Cf. A000005, A003459, A067012, A062895, A350867, A354746.

Select[Range[10000],CountDistinct[DivisorSigma[0,FromDigits /@ Permutations[IntegerDigits[#]]]]==1&&CountDistinct[IntegerDigits[#]]>1&]

from sympy import divisor_count
from itertools import permutations
def ok(n):
    s, d = str(n), divisor_count(n)
    if len(set(s)) == 1: return False
    return all(d==divisor_count(int("".join(p))) for p in permutations(s))
print([k for k in range(5500) if ok(k)]) # Michael S. Branicky, Jun 05 2022

A109093 Fully-transmutable primes: Transmutable primes such that each transmutation is itself a transmutable prime (A108388).

Original entry on oeis.org

139119131, 193113191, 319339313, 391331393, 913993919, 931991939, 1319999199391, 1913333133931, 3139999399193, 3931111311913, 9193333933139, 9391111911319, 11333911193113, 11999311139119, 33111933391331

Offset: 0

Rick L. Shepherd, Jun 18 2005

See the definitions of "transmutable" and "transmutation" in A108388. Some primes with two distinct digits, namely all terms of A083983, can be considered trivially fully-transmutable. This subsequence of A108388 considers only transmutable primes with more distinct digits. These are primes such that all permutations of assignments of their distinct digits to their shared digit pattern produces primes. (Contrast this with the absolute primes, A003459, where all permutations of the digits themselves produce primes.). Fully-transmutable primes with three distinct digits occur in sets of 3! = 6. Fully-transmutable primes with four distinct digits, if any, would occur in sets of 4! = 24 and would also be a subsequence of A108389.

			The first six terms share the digit pattern d1 d2 d3 d1 d1 d3 d1 d2 d1. Each of these terms is a (9-digit) prime corresponding to one of the 3! = 6 bijective mappings of {1,3,9} onto {d1,d2,d3}. There are no other such primes with nine or fewer digits.

Cf. A108388 (transmutable primes), A083983 (transmutable primes with two distinct digits), A108389 (transmutable primes with four distinct digits), A003459 (absolute primes), A108387 (doubly-transmutable primes).

A117639 Complete primes: prime numbers such that every permutation of its digits is prime and all the digits are primes.

Original entry on oeis.org

2, 3, 5, 7, 37, 73, 337, 373, 733

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 09 2006

			337 is a complete prime because 337 is prime and its permutations (373,733) are prime and the digits (3 and 7) are also prime.

Cf. A003459.

Select[Prime[Range[10^4]],ContainsOnly[IntegerDigits[#],{2,3,5,7}]&&AllTrue[FromDigits/@Permutations[IntegerDigits[#]],PrimeQ]&] (* James C. McMahon, Sep 25 2024 *)

A172435 Partial sums of circular primes A016114.

Original entry on oeis.org

2, 5, 10, 17, 28, 41, 58, 95, 174, 287, 484, 683, 1020, 2213, 5992, 17931, 37868, 231807, 431740, 1111111111111542851, 11112222222222222653962

Offset: 1

Jonathan Vos Post, Feb 02 2010

Circular primes are a generalization of palindromatic primes (A002385): numbers which remain prime under cyclic shifts of digits. 484 is the first square partial sum of circular primes. The subsequence of prime partial sums of circular primes begins: 2, 5, 17, 41, 683, 2213. The subsubsequence of circular prime partial sums of circular primes begins 2, 5, 17, and what is the next? What are the analogs in other bases?

			a(21) = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 37 + 79 + 113 + 197 + 199 + 337 + 1193 + 3779 + 11939 + 19937 + 193939 + 199933 + 1111111111111111111 + 11111111111111111111111.

Cf. A000040, A002385, A004023, A003459, A016114, A045978, A068652.

A185104 Primes with property that each swapping any pair of digits also gives a prime.

Original entry on oeis.org

11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 179, 199, 311, 337, 373, 733, 919, 991, 3911

Offset: 1

Jaroslav Krizek, Dec 26 2012

Conjecture: next terms are 1111111111111111111 and 11111111111111111111111.

			Prime 179 is a term because 197 and 719 are also prime.
Prime 1913 is not a term because 9113 is not prime (even though 1193, 1319, 1931 and 3911 are primes).

Cf. A003459 (absolute primes).

Subsequence of A069706 (primes with property that swapping first and last digits also gives a prime).

A225193 Composite numbers such that every non-identity permutation gives a prime.

Original entry on oeis.org

14, 16, 20, 30, 32, 34, 35, 38, 50, 70, 74, 76, 91, 92, 95, 98, 110, 118, 119, 133, 772, 775, 778, 779, 1118, 3337, 7771, 77779

Offset: 1

Jayanta Basu, May 01 2013

			772 is a member since both 727 and 277 are primes.

Cf. A003459, A068652, A068653.

t={}; Do[p=Permutations[IntegerDigits[n]]; c=Length[p]; cn=Length[Select[Table[FromDigits[k],{k,p}], PrimeQ]]; If[!PrimeQ[n] && c>1 && cn==c-1, AppendTo[t,n]], {n,10,100000}]; t

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 not isprime(k) and all((t:=int("".join(m)))==k or isprime(t) for m in permutations(s)))
print(list(islice(agen(), 28))) # Michael S. Branicky, Dec 29 2023

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

A344468 Semiprimes k such that every permutation of the digits of k is a semiprime.

Original entry on oeis.org

4, 6, 9, 15, 22, 26, 33, 39, 49, 51, 55, 58, 62, 77, 85, 93, 94, 111, 155, 177, 178, 187, 226, 262, 339, 355, 393, 515, 535, 551, 553, 622, 717, 718, 771, 781, 817, 871, 899, 933, 989, 998, 1111, 3777, 4555, 5455, 5545, 5554, 5999, 7377, 7737, 7773, 7999, 9599

Offset: 1

Ctibor O. Zizka, May 20 2021

			k = 15, A001222(15) = A001222(51) = 2, thus 15 and 51 are terms;
k = 178, A001222(178) = A001222(187) = A001222(718) = A001222(781) = A001222(817) = A001222(871) = 2, thus 178, 187, 718, 781, 817, 871 are terms.

Cf. A000005, A001222, A001358, A003459.

Subsequence of A085751 and A263106.

q[n_] := AllTrue[Permutations[IntegerDigits[n]], PrimeOmega[FromDigits[#]] == 2 &]; Select[Range[10^4], q] (* Amiram Eldar, May 20 2021 *)

A377564 Primes that contain at least two different even digits and at least two different odd digits such that any permutation of the odd digits and any permutation of the even digits produces a prime. Permutations with leading 0s are disregarded; ie. if permutations of even digits in a prime p produce a number with a leading 0 that is not prime, p is still in the sequence.

Original entry on oeis.org

1249, 1429, 1487, 1847, 2309, 2617, 2671, 2903, 4019, 4091, 6037, 6073, 6217, 6271, 6389, 6709, 6907, 6983, 7481, 7841, 8039, 8093, 8369, 8963, 9241, 9421, 20129, 20177, 20389, 20717, 20771, 20921, 20983, 21013, 21031, 22109, 22901, 23011

Offset: 1

Enrique Navarrete, Nov 01 2024

Relaxed version of A376502. For example, 2309 is not in A376502 since 329 is not prime; however, 2309 is in this sequence since nonprimes with leading 0s such as 329 that result from permutations of even digits are disregarded.

The primes in the sequence cannot contain 5.

			The primes 80107 and 80701 are in the sequence even if permutations of even digits produce 187, 781, 8107, 8701 which are numbers with leading 0s that are not prime.

Cf. A000040, A003459, A376500, A376501, A376502.

cp's OEIS Frontend

A354745 Non-repdigit numbers k such that every permutation of the digits of k has the same number of divisors.

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A109093 Fully-transmutable primes: Transmutable primes such that each transmutation is itself a transmutable prime (A108388).

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A117639 Complete primes: prime numbers such that every permutation of its digits is prime and all the digits are primes.

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A172435 Partial sums of circular primes A016114.

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A185104 Primes with property that each swapping any pair of digits also gives a prime.

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A225193 Composite numbers such that every non-identity permutation gives a 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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A344468 Semiprimes k such that every permutation of the digits of k is a semiprime.

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