A176553 - OEIS

%I #36 Aug 22 2022 14:25:21

%S 1,3,7,9,13,21,31,37,67,73,79,97,103,109,121,151,163,181,183,193,219,

%T 223,229,237,277,283,307,363,367,373,381,409,433,439,471,487,489,499,

%U 511,523,571,601,603,607,613,619,657,669,709,733,787,811,817,819,823,841,867

%N Numbers m such that concatenations of divisors of m are noncomposites.

%C 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

%C Conjecture: this sequence is a subsequence of A003136 (Loeschian numbers). - _Davide Rotondo_, Jan 02 2022

%C 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

%H Bill McEachen, <a href="/A176553/b176553.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%o (PARI) is(n)=my(d=divisors(n)); d[1]="1"; isprime(eval(concat(d))) || n==1 \\ _Charles R Greathouse IV_, Sep 23 2016

%o (Python)

%o from sympy import divisors, isprime

%o def ok(m): return m==1 or isprime(int("".join(str(d) for d in divisors(m))))

%o print([m for m in range(1, 900) if ok(m)]) # _Michael S. Branicky_, Feb 05 2022

%Y Cf. A037278, A176554, A176555.

%Y Subsequence of A045572.

%K nonn,base

%O 1,2

%A _Jaroslav Krizek_, Apr 20 2010

%E Edited and extended by _Charles R Greathouse IV_, Apr 30 2010

%E Data corrected by _Bill McEachen_, Nov 03 2021