cp's OEIS Frontend

A330706 Numbers m such that the prime factorization of m! contains no composite exponents.

%I A330706 #103 Sep 01 2020 17:56:40
%S A330706 1,2,3,4,5,8,14
%N A330706 Numbers m such that the prime factorization of m! contains no composite exponents.
%C A330706 This sequence is finite and a(7) = 14 is the last term. Nagura (see references) proves that for n >= 25, there is always a prime between n and 1.2*n. Hence, for any prime p > 25, there is always a number m between 4p and 4.8*p, and so floor(m/p) = 4. Since by assumption p > 4, floor(floor(m/p)/p) = 0 and so m! is divisible by p^4 but not p^5. It remains to check the primes up to 25 individually. - _Charles R Greathouse IV_, Apr 14 2020
%H A330706 J. Nagura, <a href="http://projecteuclid.org/euclid.pja/1195570997">On the interval containing at least one prime number</a>, Proc. Japan Acad., 28 (1952), 177-181.
%e A330706 4 is a term since 4! = (2^3)*(3^1) and the multiplicity of 2 is 3 which is prime and the multiplicity of 3 is 1.
%t A330706 Select[Range[100], !AnyTrue[FactorInteger[#!][[;;,2]], CompositeQ] &] (* _Amiram Eldar_, Mar 29 2020 *)
%o A330706 (PARI)
%o A330706 ok(n)={my(f=factor(n!)[,2]); for(i=1, #f, if(f[i]<>1 && !isprime(f[i]), return(0))); 1}
%o A330706 {select(ok, [1..100])} \\ _Andrew Howroyd_, Mar 29 2020
%o A330706 (PARI) f(m,p)=my(s); while(m\=p, s+=m); s;
%o A330706 is(n)=forprime(p=2,n\4+1, if(!isprime(f(n,p)), return(0))); 1;
%o A330706 select(is,[1..25]) \\ _Charles R Greathouse IV_, Apr 14 2020
%Y A330706 Cf. A000142, A115627.
%K A330706 nonn,fini,full
%O A330706 1,2
%A A330706 _Devansh Singh_, Mar 29 2020