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%I #12 Nov 07 2017 03:27:30
%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 3)
%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
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