A294110 - OEIS

%I #16 Nov 07 2024 15:22:33

%S 24,160,896,1215,9720,15309,22528,106496,122472,546875,1948617,

%T 2228224,9961472,15588936,17500000,20726199,132890625,165809592,

%U 192937984,537109375,1063125000,2195382771,15569256448,15869140625,17187500000,17563062168,21750594173,22082967873,66571993088,130517578125

%N Numbers with prime factorization of the form p_1^p_2*p_2^p_3*...p_(n-1)^p_n*p_n where p_(n-1) < p(n) and n > 1.

%C All members of this sequence, by definition, only have primes and 1 as exponents of prime factors.

%H Charles R Greathouse IV, <a href="/A294110/b294110.txt">Table of n, a(n) for n = 1..10000</a>

%e 24 is part of the sequence because its prime factorization is 2^3*3.

%e 122472 is part of the sequence because its prime factorization is 2^3*3^7*7

%e 10756480 is not part of the sequence because it prime factorization is 2^7*7^5*5. This does not follow the rule where each base in the chain must be greater than the previous (7<5 is not true).

%o (Python)

%o def prime_factors(n):

%o     factors = {}

%o     i = 2

%o     while n != 1:

%o         while n % i == 0:

%o             n /= i

%o             if i in factors:

%o                 factors[i] += 1

%o             else:

%o                 factors[i] = 1

%o         i += 1

%o     return factors

%o def a(n):

%o     i = 1

%o     c = 0

%o     while c < n:

%o         i += 1

%o         p = prime_factors(i)

%o         if len(p) > 1 and list(p.keys())[1:]+[1] == list(p.values()):

%o             c +=1

%o     return i

%o (PARI) is(n)=my(f=factor(n)); if(#f~<2, return(0)); for(i=2,#f~, if(f[i,1]!=f[i-1,2], return(0))); f[#f~,2]==1 \\ _Charles R Greathouse IV_, Oct 22 2017

%o (PARI) get(q,N)=my(v,pq); if(N>>q == 0, return(if(N<1,[],[1]))); v=List([1]); forprime(p=2,min(sqrtnint(N,q),q-1), pq=p^q; u=pq*get(p,N\pq); for(i=1,#u, listput(v,u[i])); u=0); Set(v)

%o list(lim)=my(v=List(),u,t); lim\=1; forprime(q=3,lambertw(log(2)*lim)\log(2), forprime(p=2,min(sqrtnint(lim,q),q-1), t=p^q*q; u=t*get(p,lim\t); for(i=1,#u, listput(v,u[i])); u=0)); Set(v) \\ _Charles R Greathouse IV_, Oct 22 2017

%Y Subsequence of A046099.

%K nonn

%O 1,1

%A _Matthew McCaskill_, Oct 22 2017

%E a(10)-a(30) from _Charles R Greathouse IV_, Oct 22 2017

%E Definition corrected by _Jens Kruse Andersen_, Oct 28 2017