A176553 Numbers m such that concatenations of divisors of m are noncomposites.

1, 3, 7, 9, 13, 21, 31, 37, 67, 73, 79, 97, 103, 109, 121, 151, 163, 181, 183, 193, 219, 223, 229, 237, 277, 283, 307, 363, 367, 373, 381, 409, 433, 439, 471, 487, 489, 499, 511, 523, 571, 601, 603, 607, 613, 619, 657, 669, 709, 733, 787, 811, 817, 819, 823, 841, 867

Offset: 1

Author

Jaroslav Krizek, Apr 20 2010

Keywords

• nonn
• base

Comments

Do all primes p > 5 have a multiple in this sequence? This holds at least for p < 10^4. - Charles R Greathouse IV, Sep 23 2016

Conjecture: this sequence is a subsequence of A003136 (Loeschian numbers). - Davide Rotondo, Jan 02 2022

If m is not in A003136, there is a prime p == 2 (mod 3) such that the exponent of p in the factorization of m is odd, then we have 3 | 1+p | 1+p+p^2+...+p^(2*r-1) | sigma(m), sigma = A000203 is the sum of divisors, so the concatenation of the divisors of m is also divisible by 3. - Jianing Song, Aug 22 2022

Examples

			a(6) = 21: the divisors of 21 are 1,3,7,21, and their concatenation 13721 is noncomposite.
		

Links

• Bill McEachen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A037278, A176554, A176555.

Subsequence of A045572.

Programs

Mathematica

Select[Range[10^3], ! CompositeQ@ FromDigits@ Flatten@ IntegerDigits@ Divisors@ # &] (* Michael De Vlieger, Sep 23 2016 *)

PARI

is(n)=my(d=divisors(n)); d[1]="1"; isprime(eval(concat(d))) || n==1 \\ Charles R Greathouse IV, Sep 23 2016

Python

from sympy import divisors, isprime
def ok(m): return m==1 or isprime(int("".join(str(d) for d in divisors(m))))
print([m for m in range(1, 900) if ok(m)]) # Michael S. Branicky, Feb 05 2022

Extensions

Edited and extended by Charles R Greathouse IV, Apr 30 2010

Data corrected by Bill McEachen, Nov 03 2021