A227815 Composite numbers n divisible by their concatenated exponents in prime factorization.

4, 16, 22, 27, 33, 55, 63, 77, 143, 187, 209, 222, 248, 253, 256, 319, 341, 407, 451, 473, 484, 517, 555, 583, 649, 656, 671, 737, 777, 781, 803, 837, 869, 913, 979, 1067, 1111, 1133, 1152, 1177, 1199, 1221, 1243, 1397, 1441, 1443, 1507, 1529, 1639, 1661, 1727

Offset

1,1

Comments

The numbers 2^(2^m), m = 1, 2,... are in the sequence. A majority of semiprimes of the form 11*p where is p prime different from 11 are in the sequence. The numbers of the form p*111 = p*3*37 where p is prime different from 3 or 37 are in the sequence. In the general case, the numbers of the form n = p_1*p_2*...*p_m*R where the p_k are prime numbers and R is a repunit number (A002275) with q digits "1" and (q-m) prime divisors are in the sequence, for example the numbers of the form n = p_1*p_2*p_3*p_4*11111111 = p_1*p_2*p_3*p_4*11*73*101*137 are in the sequence if the primes p_k are different from 11, 73, 101 or 137. So, 11111111 divides n.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

248 = 2^3*31 => 31 is the concatenate exponents 3 and 1, so 31 divides 248.

Maple

with(numtheory):for n from 1 to 10000 do:x:=ifactors(n):y:=x[2]; n1:=nops(y):s:=0:for i from 1 to n1 do:z:=y[i][2]:s:=s+z*10^(n1-i):od:if type(n, prime)=false and irem(n, s)=0 then printf(`%d, `, n):else fi:od:

Mathematica

With[{predicate = And[CompositeQ[#], Divisible[#, FromDigits[Join @@ IntegerDigits@(Last /@ FactorInteger[#])]]] &},
Select[Range[10000], predicate]] (* Sidney Cadot, Feb 19 2023 *)

Prog

(Python)
from sympy import isprime, factorint
def ok(n): return n > 1 and not isprime(n) and n%int("".join(str(e) for e in factorint(n).values())) == 0
print([k for k in range(1728) if ok(k)]) # Michael S. Branicky, Feb 19 2023

Crossrefs

Cf. A002275, A037916, A070529.

Sequence in context: A345763 A354712 A067671 * A075331 A237878 A065661

Adjacent sequences: A227812 A227813 A227814 * A227816 A227817 A227818

Keyword

nonn,base

Author

Michel Lagneau, Jul 31 2013

Status

approved