A067012 -id:A067012 - cp's OEIS frontend

A067013 Numbers with property that every permutation of digits (dropping any leading zeros) is 1 or a composite number.

1, 4, 6, 8, 9, 10, 12, 15, 18, 21, 22, 24, 25, 26, 27, 28, 33, 36, 39, 40, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 72, 75, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 93, 94, 96, 99, 100, 102, 105, 108, 111, 114, 116, 117, 120, 122

Offset: 1

Lior Manor, Dec 26 2001

			10 is a term since 10 is not a prime number and the permutation (0)1 is not a prime number either.

T. D. Noe, Table of n, a(n) for n = 1..10000

Subsequence of A248010.

Cf. A067012.

t={};Do[l1=Table[FromDigits[k],{k,Permutations[IntegerDigits[n]]}]; If[Select[l1,PrimeQ]=={},AppendTo[t,n]],{n,122}];t (* Jayanta Basu, May 03 2013 *)

A287198 Numbers with the property that every cyclic permutation of its digits is a composite number with none of its permutations sharing any common prime factors.

Original entry on oeis.org

4, 6, 8, 9, 25, 49, 52, 56, 58, 65, 85, 94, 116, 134, 145, 158, 161, 178, 187, 253, 275, 295, 325, 341, 358, 413, 451, 514, 527, 529, 532, 581, 583, 589, 611, 718, 752, 781, 815, 817, 835, 871, 895, 899, 952, 958, 989, 998, 1154, 1156, 1159, 1165, 1189, 1192

Offset: 1

Luke Zieroth, May 21 2017

Subsequence of A052382, as a number with a zero digit have cyclic permutations of the forms 0x and x0 which share prime factors of x. The only exception to this argument is 10, but 01 is not composite, so 10 is not a member of the sequence as well. - Chai Wah Wu, May 24 2017

If m is a multiple of 11 with an even number of digits, then m is not a term. - Chai Wah Wu, May 30 2017

			The numbers formed by cyclic permutations of 134 are 341 and 413. The factors of 134 are 2 and 67, the factors of 341 are 11 and 31, and the factors of 413 are 7 and 59. Since these numbers are all composite and none share any common factors with each other, 134 is included on the list.

Luke Zieroth and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 212 terms from Luke Zieroth)

Cf. A052382, A068652, A067012.

ok[n_] := Catch@ Block[{t = FromDigits /@ (RotateLeft[IntegerDigits[n], #] & /@ Range[ IntegerLength@ n])}, If[! AllTrue[t, CompositeQ], Throw@False]; Do[ If[ GCD[t[[i]], t[[j]]] > 1, Throw@False], {i, Length@t}, {j, i-1}]; True]; Select[ Range@ 1200, ok] (* Giovanni Resta, May 25 2017 *)

is(n) = {my(d=digits(n), v=vector(#d)); v[1]=n; if(isprime(n)||n==10, return(0)); for(i=2, #d, v[i] = v[i-1]\10; v[i] = v[i]+(v[i-1]-v[i]*10)*10^(#d-1); if(isprime(v[i]), return(0)); for(j=1,i-1,if(gcd(v[j], v[i])>1, return(0)))); n>1} \\ David A. Corneth, May 25 2017

from gmpy2 import is_prime, gcd, mpz
A287198_list, n  = [], 2
while n <= 10**6:
    s = str(n)
    if not is_prime(n) and '0' not in s:
        k = n
        for i in range(len(s)-1):
            s = s[1:]+s[0]
            m = mpz(s)
            if is_prime(m) or gcd(k,m) > 1:
                break
            k *= m
        else:
            A287198_list.append(n)
    n += 1 # Chai Wah Wu, May 27 2017

A091897 Numbers that remain prime or composite for all permutations of their digits.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 17, 18, 21, 22, 24, 25, 26, 27, 28, 31, 33, 36, 37, 39, 40, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 71, 72, 73, 75, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 90, 93, 94, 96, 97, 99, 102, 105

Offset: 1

Rick L. Shepherd, Feb 09 2004

Union of A003459 and A067012. The repdigit numbers (A010785) > 1 are also clearly a subsequence.

Cf. A003459 (absolute primes), A067012 ('absolute composites'), A091898 (complement of A091897), A010785 (repdigit numbers).

A248010 Non-primatic permutable numbers: All permutations of the number's digits except the ones resulting in leading zeros are nonprimes.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 33, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 93, 94, 96, 99, 100, 102, 105, 108, 111, 114, 116, 117, 120, 122, 123, 126, 129, 132, 135, 138, 141, 144, 147

Offset: 1

Andreas Boe, Sep 29 2014

This sequence differs slightly from "absolute composite numbers". 30 is not an absolute composite since 03 is counted as a prime, but in this sequence permutations with leading zeros are disqualified as viable permutations.

			7000 qualifies since it is a composite and the only allowed permutation of its four digits.

Andreas Boe, Table of n, a(n) for n = 1..10000

Absolute composites: A067012, A067013.
Monoprimatic permutable numbers: A245808.
Biprimatic permutable numbers: A246043.

A091898 Numbers that change from composite to prime or vice versa for at least one permutation of their digits.

Original entry on oeis.org

14, 16, 19, 20, 23, 29, 30, 32, 34, 35, 38, 41, 43, 47, 50, 53, 59, 61, 67, 70, 74, 76, 83, 89, 91, 92, 95, 98, 101, 103, 104, 106, 107, 109, 110, 112, 115, 118, 119, 121, 124, 125, 127, 128, 130, 133, 134, 136, 137, 139, 140, 142, 143, 145, 146, 149, 151, 152, 154

Offset: 1

Rick L. Shepherd, Feb 09 2004

This is actually a subsequence of the complement of A091897, the union of A003459 and A067012: This sequence contains no powers of 10 (A011557) as 1 is not prime.

Clearly also no repdigit number (A010785) is a term nor is any number with only even digits (except for 20,200,2000,...) nor is any number divisible by 3 (except for 30,300,3000,...). Among other primes, this sequence does include all primes p > 5 which contain at least one of the digits 0,2,4,5,6,8.

			14=2*7 (composite) is a term as a permutation of its digits gives 41 (prime).
Hence 41 is also a term. 19 (prime) is a term as 91=7*13 (composite). Thus 91
is also a term. 130=2*5*13 (composite) is a term (even though the permutation
310=2*5*31 is also composite) because another permutation (0)13 (prime) exists
(dropping the leading 0). 13, however, is not a term as 31 is also prime (13
and 31 are members of A003459).

Cf. A003459 (absolute primes), A067012 ('absolute composites'), A091897 (union of A003459 and A067012), A010785 (repdigit numbers).

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

Original entry on oeis.org

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

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A067013 Numbers with property that every permutation of digits (dropping any leading zeros) is 1 or a composite number.

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A287198 Numbers with the property that every cyclic permutation of its digits is a composite number with none of its permutations sharing any common prime factors.

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A091897 Numbers that remain prime or composite for all permutations of their digits.

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A248010 Non-primatic permutable numbers: All permutations of the number's digits except the ones resulting in leading zeros are nonprimes.

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A091898 Numbers that change from composite to prime or vice versa for at least one permutation of their digits.

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

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