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A176553 Numbers m such that concatenations of divisors of m are noncomposites. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
LINKS
EXAMPLE
a(6) = 21: the divisors of 21 are 1,3,7,21, and their concatenation 13721 is noncomposite.
MATHEMATICA
Select[Range[10^3], ! CompositeQ@ FromDigits@ Flatten@ IntegerDigits@ Divisors@ # &] (* Michael De Vlieger, Sep 23 2016 *)
PROG
(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
CROSSREFS
Subsequence of A045572.
Sequence in context: A111250 A118643 A043772 * A109370 A018663 A110575
KEYWORD
nonn,base
AUTHOR
Jaroslav Krizek, Apr 20 2010
EXTENSIONS
Edited and extended by Charles R Greathouse IV, Apr 30 2010
Data corrected by Bill McEachen, Nov 03 2021
STATUS
approved

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Last modified July 13 02:50 EDT 2024. Contains 374265 sequences. (Running on oeis4.)