A294110 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.

24, 160, 896, 1215, 9720, 15309, 22528, 106496, 122472, 546875, 1948617, 2228224, 9961472, 15588936, 17500000, 20726199, 132890625, 165809592, 192937984, 537109375, 1063125000, 2195382771, 15569256448, 15869140625, 17187500000, 17563062168, 21750594173, 22082967873, 66571993088, 130517578125

Offset

1,1

Comments

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

24 is part of the sequence because its prime factorization is 2^3*3.
122472 is part of the sequence because its prime factorization is 2^3*3^7*7
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).

Prog

(Python 3)
def prime_factors(n):
    factors = {}
    i = 2
    while n != 1:
        while n % i == 0:
            n /= i
            if i in factors:
                factors[i] += 1
            else:
                factors[i] = 1
        i += 1
    return factors
def a(n):
    i = 1
    c = 0
    while c < n:
        i += 1
        p = prime_factors(i)
        if len(p) > 1 and list(p.keys())[1:]+[1] == list(p.values()):
            c +=1
    return i
(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
(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)
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

Crossrefs

Subsequence of A046099.

Sequence in context: A231341 A186862 A305165 * A136380 A250323 A250142

Adjacent sequences: A294107 A294108 A294109 * A294111 A294112 A294113

Keyword

nonn

Author

Matthew McCaskill, Oct 22 2017

Extensions

a(10)-a(30) from Charles R Greathouse IV, Oct 22 2017

Definition corrected by Jens Kruse Andersen, Oct 28 2017

Status

approved